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Properties

Label 3.3.961.1-4.1-a3
Base field 3.3.961.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.961.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 10 x + 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -10, -1, 1]))
 
gp: K = nfinit(Polrev([8, -10, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -10, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-\frac{6151408334958753821769050847112196849}{2}a^{2}-\frac{14130218892584489931013217519954906083}{2}a+7462877426720882640138607519245152578\right){x}+\frac{16618716709396809372366318063381491518225891025259152029}{2}a^{2}+\frac{38174364638273178291400194444212921470457543144134251711}{2}a-20161796947667300676602389656879723718330935346799904418\)
sage: E = EllipticCurve([K([1,0,0]),K([-1,1,0]),K([-2,-1/2,1/2]),K([7462877426720882640138607519245152578,-14130218892584489931013217519954906083/2,-6151408334958753821769050847112196849/2]),K([-20161796947667300676602389656879723718330935346799904418,38174364638273178291400194444212921470457543144134251711/2,16618716709396809372366318063381491518225891025259152029/2])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([-1,1,0]),Polrev([-2,-1/2,1/2]),Polrev([7462877426720882640138607519245152578,-14130218892584489931013217519954906083/2,-6151408334958753821769050847112196849/2]),Polrev([-20161796947667300676602389656879723718330935346799904418,38174364638273178291400194444212921470457543144134251711/2,16618716709396809372366318063381491518225891025259152029/2])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![-1,1,0],K![-2,-1/2,1/2],K![7462877426720882640138607519245152578,-14130218892584489931013217519954906083/2,-6151408334958753821769050847112196849/2],K![-20161796947667300676602389656879723718330935346799904418,38174364638273178291400194444212921470457543144134251711/2,16618716709396809372366318063381491518225891025259152029/2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2-2a+2)\) = \((-1/2a^2-1/2a+3)\cdot(-1/2a^2+1/2a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a+4)\) = \((-1/2a^2-1/2a+3)\cdot(-1/2a^2+1/2a+4)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32 \) = \(2\cdot2^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{212068284953}{32} a^{2} + \frac{487135796771}{32} a - \frac{257280852307}{16} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{1012047966960026891}{4} a^{2} + \frac{1162372787193307213}{2} a - \frac{2455629513151280341}{4} : -\frac{1012047966960026893}{8} a^{2} - 290593196798326803 a + \frac{2455629513151280349}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 101.98608543974520140521768089278751447 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.6449368619313742162131884014965728140 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((-1/2a^2-1/2a+3)\) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-1/2a^2+1/2a+4)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 4.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.