| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 10.1-a1 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{26} \cdot 5^{7} \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.385680830$ |
1.742129368 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -67 a + 360\) , \( 905 a + 1396\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(-67a+360\right){x}+905a+1396$ |
| 10.1-a2 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{14} \cdot 5 \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$9.699765812$ |
1.742129368 |
\( \frac{2911}{20} a - \frac{193027}{20} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -2 a\) , \( -a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2-2a{x}-a+4$ |
| 10.1-a3 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{2} \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$9.699765812$ |
1.742129368 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a - 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(a-3\right){x}+1$ |
| 10.1-a4 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{7} \cdot 5^{14} \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.385680830$ |
1.742129368 |
\( \frac{379001974281391}{781250000000} a - \frac{103373105456887}{781250000000} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -4 a + 17\) , \( 12 a - 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a+17\right){x}+12a-1$ |
| 10.4-a1 |
10.4-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{26} \cdot 5^{7} \) |
$0.88475$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.385680830$ |
1.742129368 |
\( \frac{780929100411181}{1280000000} a - \frac{336050998176533}{160000000} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 62 a + 289\) , \( -550 a + 1785\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(62a+289\right){x}-550a+1785$ |
| 10.4-a2 |
10.4-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{14} \cdot 5 \) |
$0.88475$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$9.699765812$ |
1.742129368 |
\( -\frac{2911}{20} a - \frac{47529}{5} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -3 a - 6\) , \( -4 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-3a-6\right){x}-4a+7$ |
| 10.4-a3 |
10.4-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{2} \) |
$0.88475$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$9.699765812$ |
1.742129368 |
\( \frac{19951}{50} a - \frac{33947}{25} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a - 2\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(a-2\right){x}-1$ |
| 10.4-a4 |
10.4-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{7} \cdot 5^{14} \) |
$0.88475$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.385680830$ |
1.742129368 |
\( -\frac{379001974281391}{781250000000} a + \frac{34453608603063}{97656250000} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 6 a + 13\) , \( -7 a + 24\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(6a+13\right){x}-7a+24$ |
| 14.2-a1 |
14.2-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{9} \) |
$0.96239$ |
$(2,a), (7,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$1.122170831$ |
0.806191324 |
\( -\frac{3429643149944533}{10578455953408} a - \frac{436912252725523}{10578455953408} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a + 29\) , \( -24 a - 42\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a+29\right){x}-24a-42$ |
| 14.2-a2 |
14.2-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{54} \cdot 7^{3} \) |
$0.96239$ |
$(2,a), (7,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.374056943$ |
0.806191324 |
\( \frac{1524255431343666912883}{6178938688752320512} a + \frac{1865190650273146662373}{6178938688752320512} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 4 a - 266\) , \( 467 a + 1516\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a-266\right){x}+467a+1516$ |
| 14.2-a3 |
14.2-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7 \) |
$0.96239$ |
$(2,a), (7,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$10.09953748$ |
0.806191324 |
\( \frac{17387}{28} a + \frac{61997}{28} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a-1\right){x}$ |
| 14.2-a4 |
14.2-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{3} \) |
$0.96239$ |
$(2,a), (7,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.366512495$ |
0.806191324 |
\( -\frac{25508811437}{21952} a + \frac{14931496453}{21952} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 4 a + 4\) , \( -a - 38\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a+4\right){x}-a-38$ |
| 14.3-a1 |
14.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{3} \) |
$0.96239$ |
$(2,a+1), (7,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.366512495$ |
0.806191324 |
\( \frac{25508811437}{21952} a - \frac{1322164373}{2744} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -5 a + 9\) , \( a - 39\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+9\right){x}+a-39$ |
| 14.3-a2 |
14.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{9} \) |
$0.96239$ |
$(2,a+1), (7,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$1.122170831$ |
0.806191324 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 29\) , \( 24 a - 66\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+29{x}+24a-66$ |
| 14.3-a3 |
14.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{54} \cdot 7^{3} \) |
$0.96239$ |
$(2,a+1), (7,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.374056943$ |
0.806191324 |
\( -\frac{1524255431343666912883}{6178938688752320512} a + \frac{423680760202101696907}{772367336094040064} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -5 a - 261\) , \( -467 a + 1983\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a-261\right){x}-467a+1983$ |
| 14.3-a4 |
14.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7 \) |
$0.96239$ |
$(2,a+1), (7,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$10.09953748$ |
0.806191324 |
\( -\frac{17387}{28} a + \frac{19846}{7} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-{x}$ |
| 20.3-a1 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{27} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.463053980$ |
$2.448298200$ |
0.814469976 |
\( \frac{4201360591}{8000000} a + \frac{36727033713}{8000000} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 4 a + 40\) , \( -38 a + 48\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(4a+40\right){x}-38a+48$ |
| 20.3-a2 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{17} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.389161942$ |
$7.344894602$ |
0.814469976 |
\( -\frac{240871}{200} a + \frac{1256497}{200} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2-a{x}$ |
| 20.3-a3 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{19} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.778323884$ |
$7.344894602$ |
0.814469976 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -4 a - 4\) , \( 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-4a-4\right){x}+16$ |
| 20.3-a4 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{33} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$0.926107961$ |
$2.448298200$ |
0.814469976 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 11 a + 21\) , \( -5 a + 45\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(11a+21\right){x}-5a+45$ |
| 20.3-b1 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{30} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.138318178$ |
1.226687881 |
\( \frac{5721159718441}{512000} a - \frac{17013218122889}{64000} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 78 a - 963\) , \( -1718 a + 12067\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(78a-963\right){x}-1718a+12067$ |
| 20.3-b2 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{29} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( \frac{94333009}{12800} a - \frac{23122663}{12800} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 3 a + 29\) , \( 16 a - 58\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(3a+29\right){x}+16a-58$ |
| 20.3-b3 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{34} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( \frac{7985051}{1310720} a - \frac{205461507}{1310720} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -2 a - 3\) , \( -6 a + 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-2a-3\right){x}-6a+35$ |
| 20.3-b4 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{39} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.138318178$ |
1.226687881 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 68 a - 171\) , \( 452 a - 346\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(68a-171\right){x}+452a-346$ |
| 20.4-a1 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{17} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.389161942$ |
$7.344894602$ |
0.814469976 |
\( \frac{240871}{200} a + \frac{507813}{100} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 1\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}-a$ |
| 20.4-a2 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{27} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.463053980$ |
$2.448298200$ |
0.814469976 |
\( -\frac{4201360591}{8000000} a + \frac{639506161}{125000} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -6 a + 44\) , \( 37 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-6a+44\right){x}+37a+10$ |
| 20.4-a3 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{19} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.778323884$ |
$7.344894602$ |
0.814469976 |
\( -\frac{566667}{320} a + \frac{339431}{40} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+a{x}$ |
| 20.4-a4 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{33} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$0.926107961$ |
$2.448298200$ |
0.814469976 |
\( \frac{55081295373}{32768000} a + \frac{46077139211}{4096000} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -14 a + 40\) , \( 30 a + 144\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-14a+40\right){x}+30a+144$ |
| 20.4-b1 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{30} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.138318178$ |
1.226687881 |
\( -\frac{5721159718441}{512000} a - \frac{130384585264671}{512000} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -80 a - 883\) , \( 1717 a + 10350\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-80a-883\right){x}+1717a+10350$ |
| 20.4-b2 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{39} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.138318178$ |
1.226687881 |
\( \frac{1520638086962303}{262144000000} a - \frac{817791717649607}{262144000000} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -70 a - 103\) , \( -453 a + 106\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-70a-103\right){x}-453a+106$ |
| 20.4-b3 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{34} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3\) , \( 5 a + 30\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2-3{x}+5a+30$ |
| 20.4-b4 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{29} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( -\frac{94333009}{12800} a + \frac{35605173}{6400} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -5 a + 32\) , \( -17 a - 42\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-5a+32\right){x}-17a-42$ |
| 32.2-a1 |
32.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{19} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( \frac{514073}{32} a - 48120 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -2 a + 8\) , \( 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-2a+8\right){x}+4$ |
| 32.2-a2 |
32.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{31} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( -\frac{514073}{32} a - \frac{1025767}{32} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 2 a + 32\) , \( -20 a + 16\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(2a+32\right){x}-20a+16$ |
| 32.2-a3 |
32.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{35} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( \frac{85169}{1024} a + \frac{12167}{128} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 8\) , \( -4 a + 16\bigr] \) |
${y}^2={x}^3-{x}^2+8{x}-4a+16$ |
| 32.2-a4 |
32.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{35} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( -\frac{85169}{1024} a + \frac{182505}{1024} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 8\) , \( -4 a - 12\bigr] \) |
${y}^2={x}^3+{x}^2+8{x}-4a-12$ |
| 32.3-a1 |
32.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.3 |
\( 2^{5} \) |
\( 2^{20} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2Cn, 5S4 |
$1$ |
\( 2 \) |
$0.139158379$ |
$8.191609121$ |
1.637901283 |
\( 1536 a + 1280 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -a + 3\) , \( -3\bigr] \) |
${y}^2={x}^3-{x}^2+\left(-a+3\right){x}-3$ |
| 32.4-a1 |
32.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.4 |
\( 2^{5} \) |
\( 2^{20} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2Cn, 5S4 |
$1$ |
\( 2 \) |
$0.139158379$ |
$8.191609121$ |
1.637901283 |
\( -1536 a + 2816 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( a + 2\) , \( -3\bigr] \) |
${y}^2={x}^3-{x}^2+\left(a+2\right){x}-3$ |
| 32.5-a1 |
32.5-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{31} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( \frac{514073}{32} a - 48120 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -4 a + 34\) , \( 19 a - 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-4a+34\right){x}+19a-4$ |
| 32.5-a2 |
32.5-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{19} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( -\frac{514073}{32} a - \frac{1025767}{32} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 6\) , \( -a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+6{x}-a+4$ |
| 32.5-a3 |
32.5-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{35} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( \frac{85169}{1024} a + \frac{12167}{128} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 8\) , \( 4 a - 16\bigr] \) |
${y}^2={x}^3+{x}^2+8{x}+4a-16$ |
| 32.5-a4 |
32.5-a |
$4$ |
$10$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{35} \) |
$1.18333$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.684514436$ |
1.323516656 |
\( -\frac{85169}{1024} a + \frac{182505}{1024} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 8\) , \( 4 a + 12\bigr] \) |
${y}^2={x}^3-{x}^2+8{x}+4a+12$ |
| 49.1-a1 |
49.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
49.1 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.31634$ |
$(7,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \) |
$1$ |
$3.185341100$ |
2.288416601 |
\( \frac{38637}{343} a + \frac{563544}{343} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 2 a + 2\) , \( -2 a + 7\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(2a+2\right){x}-2a+7$ |
| 49.3-a1 |
49.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
49.3 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.31634$ |
$(7,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \) |
$1$ |
$3.185341100$ |
2.288416601 |
\( -\frac{38637}{343} a + \frac{602181}{343} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -2 a - 4\) , \( -a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+{x}^2+\left(-2a-4\right){x}-a+2$ |
| 50.1-a1 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{26} \cdot 5^{13} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.619695306$ |
1.558207877 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -609 a + 1079\) , \( -674 a + 33097\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-609a+1079\right){x}-674a+33097$ |
| 50.1-a2 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{14} \cdot 5^{7} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.337867144$ |
1.558207877 |
\( \frac{2911}{20} a - \frac{193027}{20} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( a - 11\) , \( -9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(a-11\right){x}-9$ |
| 50.1-a3 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{13} \cdot 5^{8} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.337867144$ |
1.558207877 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( a + 8\) , \( -8 a\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(a+8\right){x}-8a$ |
| 50.1-a4 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{19} \cdot 5^{20} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.619695306$ |
1.558207877 |
\( \frac{379001974281391}{781250000000} a - \frac{103373105456887}{781250000000} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -144 a - 112\) , \( 1415 a - 3512\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-144a-112\right){x}+1415a-3512$ |
| 50.6-a1 |
50.6-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.6 |
\( 2 \cdot 5^{2} \) |
\( 2^{26} \cdot 5^{13} \) |
$1.32301$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.619695306$ |
1.558207877 |
\( \frac{780929100411181}{1280000000} a - \frac{336050998176533}{160000000} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 605 a + 472\) , \( 1753 a + 27576\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(605a+472\right){x}+1753a+27576$ |
| 50.6-a2 |
50.6-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.6 |
\( 2 \cdot 5^{2} \) |
\( 2^{14} \cdot 5^{7} \) |
$1.32301$ |
$(2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.337867144$ |
1.558207877 |
\( -\frac{2911}{20} a - \frac{47529}{5} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -5 a - 8\) , \( -11 a + 24\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-5a-8\right){x}-11a+24$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
