LMFDB - Abelian variety search results

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Base field	Base char.	Dimension	$p$-rank	Initial coefficients

Simple	Geom. simple	Primitive	Princ. polarizable	Jacobian

Slopes	Points on variety	Points on curve	Simple factors

Newton elevation	# Jacobians	# Hyp. Jacobians	# twists	Max twist degree

Angle rank	Angle corank	End. degree	$p$-corank	(Geom.) Sq.free

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Label	Dimension	Base field	Base char.	Simple	Geom. simple	Primitive	Ordinary	Almost ordinary	Princ. polarizable	Jacobian	L-polynomial	Newton slopes	Newton elevation	$p$-rank	$p$-corank	Angle rank	Angle corank	$\mathbb{F}_q$ points on curve	$\mathbb{F}_{q^k}$ points on curve	$\mathbb{F}_q$ points on variety	$\mathbb{F}_{q^k}$ points on variety	Jacobians	Hyperelliptic Jacobians	Num. twists	Max. twist degree	End. degree	Number fields	Galois groups	Isogeny factors
4.2.ae_j_ar_bb	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 34 x^{5} + 36 x^{6} - 32 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$-1$	$[-1, 7, 2, 11, 44, 70, 125, 235, 425, 1002]$	$2$	$[2, 352, 1682, 42944, 1475342, 18057952, 267337072, 3936504704, 57996253142, 1075252855072]$	$0$	$0$	$2$	$2$	$1$	8.0.316180889.1	$C_2 \wr S_4$	simple
4.2.ae_k_at_bd	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 4 x + 10 x^{2} - 19 x^{3} + 29 x^{4} - 38 x^{5} + 40 x^{6} - 32 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$-1$	$[-1, 9, 8, 13, 29, 42, 118, 285, 494, 1039]$	$3$	$[3, 567, 3033, 54999, 935733, 12037977, 245920503, 4821927327, 66384948249, 1114354136637]$	$0$	$0$	$2$	$2$	$1$	8.0.371495353.1	$C_2 \wr S_4$	simple
4.2.ae_k_as_bb	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$-1$	$[-1, 9, 11, 21, 29, 33, 139, 349, 569, 1069]$	$4$	$[4, 736, 4228, 100096, 942524, 10113376, 292880396, 6125875200, 76563063556, 1146508814176]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.679024.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ad_f_ah
4.2.ae_l_av_bh	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$-1$	$[-1, 11, 14, 19, 34, 50, 139, 363, 599, 926]$	$6$	$[6, 1224, 7182, 90576, 1032306, 13186152, 290092368, 6457887648, 81498757662, 998409200184]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.465831.1	$C_2$, $A_4\times C_2$	1.2.ab $\times$ 3.2.ad_g_aj
4.2.ad_c_e_ak	$4$	$\F_{2}$	$2$	✓	✓	✓			✓	✓	$1 - 3 x + 2 x^{2} + 4 x^{3} - 10 x^{4} + 8 x^{5} + 8 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},1]$	$3$	$1$	$3$	$4$	$0$	$0$	$[0, 0, 12, 8, 30, 84, 154, 288, 426, 1040]$	$2$	$[2, 64, 6638, 40576, 966842, 23153344, 328442578, 4903609600, 57992641394, 1115689259584]$	$2$	$0$	$2$	$2$	$1$	8.0.391881616.1	$S_4\times C_2$	simple
4.2.ad_c_f_an	$4$	$\F_{2}$	$2$	✓	✓	✓	✓				$1 - 3 x + 2 x^{2} + 5 x^{3} - 13 x^{4} + 10 x^{5} + 8 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 0, 15, 8, 30, 57, 140, 240, 411, 920]$	$2$	$[2, 52, 7736, 37024, 905042, 15286336, 295877794, 4052054656, 56404081112, 992564991652]$	$0$	$0$	$2$	$2$	$1$	8.0.903363136.2	$C_2^2 \wr C_2$	simple
4.2.ad_d_ac_d	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 3 x^{2} - 2 x^{3} + 3 x^{4} - 4 x^{5} + 12 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 2, 3, 14, 25, 107, 147, 246, 570, 1047]$	$2$	$[2, 136, 2762, 52496, 824762, 31740904, 313875088, 4128390432, 77127764534, 1122293241976]$	$1$	$0$	$2$	$2$	$1$	8.0.1794086993.1	$C_2 \wr S_4$	simple
4.2.ad_d_ab_a	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓	✓	$1 - 3 x + 3 x^{2} - x^{3} - 2 x^{5} + 12 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$0$	$[0, 2, 6, 14, 20, 98, 126, 222, 456, 942]$	$2$	$[2, 124, 3482, 50096, 675422, 27417268, 265128502, 3783650688, 61678201562, 1013863806604]$	$1$	$0$	$2$	$2$	$1$	8.0.5198736512.1	$C_2 \wr S_4$	simple
4.2.ad_d_a_ad	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 3 x^{2} - 3 x^{4} + 12 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 2, 9, 14, 15, 83, 105, 214, 378, 887]$	$2$	$[2, 112, 4286, 49504, 560102, 21841456, 220669808, 3637256896, 52620494942, 961704095632]$	$1$	$0$	$2$	$2$	$1$	8.0.1262878369.2	$C_2^3:S_4$	simple
4.2.ad_d_b_af	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 3 x^{2} + x^{3} - 5 x^{4} + 2 x^{5} + 12 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 2, 12, 18, 25, 98, 140, 290, 444, 997]$	$3$	$[3, 153, 7461, 75429, 811248, 28538325, 291585633, 4904770725, 59868862101, 1069757042688]$	$0$	$0$	$2$	$2$	$1$	8.0.7283769025.1	$(((C_4 \times C_2): C_2):C_2):C_2$	simple
4.2.ad_d_c_ah	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 3 x^{2} + 2 x^{3} - 7 x^{4} + 4 x^{5} + 12 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 2, 15, 22, 35, 107, 161, 326, 438, 947]$	$4$	$[4, 184, 11428, 101936, 1076284, 32066968, 338845952, 5625440096, 59366597236, 1018946046184]$	$1$	$0$	$2$	$2$	$1$	8.0.316180889.1	$C_2 \wr S_4$	simple
4.2.ad_e_ae_f	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 4 x^{2} - 4 x^{3} + 5 x^{4} - 8 x^{5} + 16 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 4, 6, 20, 25, 94, 147, 196, 510, 1019]$	$3$	$[3, 243, 3573, 71199, 819843, 26915409, 316649343, 3386865231, 68101204923, 1092466212633]$	$1$	$0$	$2$	$2$	$1$	8.0.15476927681.1	$C_2 \wr S_4$	simple
4.2.ad_e_ac_ab	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 4 x^{2} - 2 x^{3} - x^{4} - 4 x^{5} + 16 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 4, 12, 20, 5, 58, 105, 244, 516, 959]$	$3$	$[3, 207, 5679, 78039, 420333, 15282189, 221488977, 4118664303, 69445759401, 1029634686477]$	$0$	$0$	$2$	$2$	$1$	8.0.5501270057.1	$C_2 \wr S_4$	simple
4.2.ad_f_ai_n	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 5 x^{2} - 8 x^{3} + 13 x^{4} - 16 x^{5} + 20 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 6, 3, 22, 45, 87, 161, 214, 516, 1151]$	$4$	$[4, 424, 2308, 82256, 1622564, 23730856, 347017696, 3614164128, 69086386444, 1240232754424]$	$1$	$0$	$2$	$2$	$1$	8.0.6419119697.1	$C_2 \wr S_4$	simple
4.2.ad_f_ah_j	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 5 x^{2} - 7 x^{3} + 9 x^{4} - 14 x^{5} + 20 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 6, 6, 18, 15, 54, 126, 162, 474, 1101]$	$3$	$[3, 297, 2547, 58509, 623568, 14372721, 259450341, 2957805477, 63893423079, 1186265786112]$	$1$	$1$	$2$	$2$	$1$	8.0.8394138297.1	$C_2 \wr S_4$	simple
4.2.ad_f_ah_l	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 5 x^{2} - 7 x^{3} + 11 x^{4} - 14 x^{5} + 20 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 6, 6, 26, 45, 102, 182, 210, 492, 1061]$	$5$	$[5, 505, 3695, 103525, 1562000, 29482405, 406501555, 3558050725, 66167211155, 1138410592000]$	$1$	$1$	$2$	$2$	$1$	8.0.1077480625.1	$(((C_4 \times C_2): C_2):C_2):C_2$	simple
4.2.ad_f_ag_h	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 5 x^{2} - 6 x^{3} + 7 x^{4} - 12 x^{5} + 20 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 6, 9, 22, 15, 63, 147, 214, 540, 1111]$	$4$	$[4, 376, 4204, 84976, 626044, 16992568, 306936928, 3641221600, 72417351484, 1197029934376]$	$0$	$0$	$2$	$2$	$1$	8.0.17603938393.1	$C_2 \wr S_4$	simple
4.2.ad_f_ae_d	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + x^{2} + x^{3} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 6, 15, 30, 15, 63, 147, 294, 636, 1011]$	$6$	$[6, 504, 9198, 158256, 588786, 16225272, 308700048, 4948032096, 87276262374, 1083180183624]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.2369943.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ac_b_b
4.2.ad_g_al_r	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 6 x^{2} - 11 x^{3} + 17 x^{4} - 22 x^{5} + 24 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 8, 3, 16, 40, 53, 112, 224, 507, 1208]$	$4$	$[4, 496, 1744, 57536, 1418764, 13840384, 237171764, 3783797504, 68073899056, 1314348644656]$	$1$	$1$	$2$	$2$	$1$	8.0.440605625.1	$C_4\wr C_2$	simple
4.2.ad_g_al_t	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 8, 3, 24, 70, 89, 140, 272, 471, 928]$	$6$	$[6, 756, 2232, 102816, 3176886, 23623488, 294556422, 4593407616, 63211518936, 1001119377636]$	$0$	$0$	$4$	$2$	$1$	4.0.1088.2, 4.0.2873.1	$D_{4}$, $D_{4}$	2.2.ac_d $\times$ 2.2.ab_b
4.2.ad_g_ak_o	$4$	$\F_{2}$	$2$	✓	✓	✓			✓		$1 - 3 x + 6 x^{2} - 10 x^{3} + 14 x^{4} - 20 x^{5} + 24 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},1]$	$3$	$1$	$3$	$4$	$0$	$0$	$[0, 8, 6, 16, 20, 44, 140, 224, 492, 1208]$	$4$	$[4, 472, 2356, 57584, 769924, 11954344, 292331708, 3819086048, 66043337812, 1313796773752]$	$0$	$0$	$2$	$2$	$1$	8.0.2189210768.1	$C_2 \wr S_4$	simple
4.2.ad_g_ak_p	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 6 x^{2} - 10 x^{3} + 15 x^{4} - 20 x^{5} + 24 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 8, 6, 20, 35, 62, 147, 228, 510, 1263]$	$5$	$[5, 595, 2735, 74375, 1249775, 15947785, 307498735, 3834849375, 68413456835, 1383274715725]$	$0$	$0$	$2$	$2$	$1$	8.0.13927884761.1	$C_2 \wr S_4$	simple
4.2.ad_g_aj_m	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓		$1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 18 x^{5} + 24 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$0$	$[0, 8, 9, 20, 15, 47, 168, 268, 540, 1163]$	$5$	$[5, 565, 3665, 81925, 643775, 12838495, 361120520, 4522341925, 72266906135, 1257133562575]$	$0$	$0$	$2$	$2$	$1$	8.0.6305297416.1	$C_2 \wr S_4$	simple
4.2.ad_g_ai_k	$4$	$\F_{2}$	$2$			✓			✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - 2 x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},1]$	$3$	$1$	$3$	$4$	$0$	$0$	$[0, 8, 12, 24, 10, 44, 182, 320, 606, 1048]$	$6$	$[6, 648, 5418, 115344, 516846, 12288024, 407025534, 5433394464, 82531651734, 1124257890888]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.1142512.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ac_c_ac
4.2.ad_g_ai_l	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 6 x^{2} - 8 x^{3} + 11 x^{4} - 16 x^{5} + 24 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 8, 12, 28, 25, 62, 175, 260, 534, 1103]$	$7$	$[7, 763, 5929, 131999, 847777, 16156525, 374139647, 4359266975, 71715589099, 1186607185813]$	$0$	$0$	$2$	$2$	$1$	8.0.4968649025.1	$C_2 \wr C_2\wr C_2$	simple
4.2.ad_g_ah_j	$4$	$\F_{2}$	$2$			✓	✓		✓	✓	$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 8, 15, 32, 20, 53, 168, 288, 591, 1008]$	$8$	$[8, 832, 8288, 176384, 695288, 13791232, 362587912, 4832216064, 80263834784, 1082552291392]$	$1$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.2464727.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ac_c_ab
4.2.ad_h_an_t	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 7 x^{2} - 13 x^{3} + 19 x^{4} - 26 x^{5} + 28 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 10, 6, 10, 25, 46, 140, 290, 510, 1125]$	$5$	$[5, 685, 2405, 44525, 864400, 12190945, 300014405, 4886039925, 68227596515, 1214544236800]$	$1$	$1$	$2$	$2$	$1$	8.0.2171225457.1	$C_2 \wr S_4$	simple
4.2.ad_h_am_r	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 5 x^{3} + 6 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 10, 9, 14, 15, 43, 189, 342, 486, 1015]$	$6$	$[6, 792, 3402, 64944, 588786, 12124728, 429411408, 5986408032, 65131864938, 1086599852712]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.3194271.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ac_d_af
4.2.ad_h_am_t	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 7 x^{2} - 12 x^{3} + 19 x^{4} - 24 x^{5} + 28 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 10, 9, 22, 45, 67, 147, 262, 486, 1155]$	$8$	$[8, 1072, 3992, 87904, 1620648, 16582768, 307764416, 4382717632, 65267168072, 1248926450832]$	$0$	$0$	$2$	$2$	$1$	8.0.3339178657.1	$C_2 \wr S_4$	simple
4.2.ad_h_al_r	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 3 x + 7 x^{2} - 11 x^{3} + 17 x^{4} - 22 x^{5} + 28 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 10, 12, 26, 35, 58, 168, 274, 462, 1085]$	$9$	$[9, 1161, 5373, 117261, 1150704, 14555457, 359213787, 4652095653, 62066322333, 1166151050496]$	$1$	$1$	$2$	$2$	$1$	8.0.1903434953.1	$C_2 \wr S_4$	simple
4.2.ad_h_ak_p	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 3 x^{3} + 6 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 10, 15, 30, 25, 43, 175, 310, 510, 975]$	$10$	$[10, 1240, 7210, 161200, 798050, 11622520, 386472880, 5295420000, 68498799670, 1046087132200]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.2256319.1	$C_2$, $A_4\times C_2$	1.2.ab $\times$ 3.2.ac_d_ad
4.2.ad_i_ap_z	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 12, 9, 16, 60, 81, 84, 256, 621, 1012]$	$10$	$[10, 1540, 4960, 61600, 2422750, 21387520, 185411530, 4334915200, 84748241440, 1086104288500]$	$0$	$0$	$4$	$2$	$1$	4.0.1088.2, 4.0.1025.1	$D_{4}$, $D_{4}$	2.2.ac_d $\times$ 2.2.ab_d
4.2.ad_i_ao_w	$4$	$\F_{2}$	$2$			✓			✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + 4 x^{2} - 6 x^{3} + 8 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},1]$	$3$	$1$	$3$	$4$	$0$	$0$	$[0, 12, 12, 16, 30, 60, 182, 352, 462, 872]$	$10$	$[10, 1480, 5530, 74000, 880550, 15550360, 404976190, 6203124000, 62423032630, 947566899400]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.503792.1	$C_2$, $S_4\times C_2$	1.2.ab $\times$ 3.2.ac_e_ag
4.2.ad_i_ao_x	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓		$1 - 3 x + 8 x^{2} - 14 x^{3} + 23 x^{4} - 28 x^{5} + 32 x^{6} - 24 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 12, 12, 20, 45, 66, 133, 276, 480, 1007]$	$11$	$[11, 1639, 5819, 80311, 1528901, 16473589, 280288789, 4639486159, 64451424389, 1080029426509]$	$0$	$0$	$2$	$2$	$1$	8.0.404205625.1	$C_2 \wr C_2\wr C_2$	simple
4.2.ad_j_aq_bb	$4$	$\F_{2}$	$2$			✓	✓		✓		$( 1 - x + 2 x^{2} )( 1 - 2 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 8 x^{5} + 8 x^{6} )$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$0$	$[0, 14, 15, 14, 35, 71, 147, 294, 456, 819]$	$14$	$[14, 2296, 8918, 59696, 1013474, 19013176, 308211568, 4978288224, 61449238214, 897698784136]$	$0$	$0$	$4$	$2$	$1$	$\Q(\sqrt{-7}) $, 6.0.400967.1	$C_2$, $A_4\times C_2$	1.2.ab $\times$ 3.2.ac_f_ah
4.2.ac_ab_d_ab	$4$	$\F_{2}$	$2$	✓	✓	✓	✓				$1 - 2 x - x^{2} + 3 x^{3} - x^{4} + 6 x^{5} - 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, -1, 4, 3, 46, 74, 113, 283, 535, 1034]$	$2$	$[2, 40, 2402, 29200, 1576762, 19648360, 236078096, 4793063200, 71868351626, 1111774886200]$	$0$	$0$	$2$	$2$	$1$	8.0.1262878369.1	$C_2^3:S_4$	simple
4.2.ac_a_b_b	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{3} + x^{4} + 2 x^{5} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 1, 4, 13, 41, 94, 92, 285, 508, 991]$	$3$	$[3, 99, 2763, 54351, 1337853, 26533089, 200157009, 4833380079, 68076554031, 1064921623029]$	$1$	$0$	$2$	$2$	$1$	8.0.20885741361.1	$C_2^3:S_4$	simple
4.2.ac_a_c_ac	$4$	$\F_{2}$	$2$	✓	✓	✓			✓	✓	$1 - 2 x + 2 x^{3} - 2 x^{4} + 4 x^{5} - 16 x^{7} + 16 x^{8}$	$[\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},\frac{3}{4},\frac{3}{4},\frac{3}{4}]$	$4$	$0$	$4$	$4$	$0$	$1$	$[1, 1, 7, 9, 41, 85, 85, 241, 457, 1041]$	$3$	$[3, 81, 3627, 38961, 1362273, 23209173, 192292341, 4050736209, 61409328201, 1117087018641]$	$3$	$0$	$2$	$2$	$1$	8.0.6598637824.2	$C_2^3:S_4$	simple
4.2.ac_a_d_ae	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓	✓	$1 - 2 x + 3 x^{3} - 4 x^{4} + 6 x^{5} - 16 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$1$	$[1, 1, 10, 9, 51, 94, 127, 257, 514, 1121]$	$4$	$[4, 88, 5872, 40304, 1923724, 26870272, 269946436, 4311641312, 68726187472, 1204947612088]$	$2$	$0$	$2$	$2$	$1$	8.0.18731962568.1	$C_2 \wr S_4$	simple
4.2.ac_a_e_ah	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + 4 x^{3} - 7 x^{4} + 8 x^{5} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 1, 13, 5, 51, 73, 141, 189, 535, 1061]$	$4$	$[4, 64, 7324, 27136, 1878284, 19100992, 300655604, 3282804736, 71768952628, 1138119893824]$	$2$	$0$	$2$	$2$	$1$	8.0.391881616.1	$S_4\times C_2$	simple
4.2.ac_b_ad_i	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓		$1 - 2 x + x^{2} - 3 x^{3} + 8 x^{4} - 6 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$1$	$[1, 3, -2, 23, 36, 60, 106, 215, 493, 848]$	$3$	$[3, 171, 999, 89091, 1063743, 15887097, 227074872, 3632863707, 66049495389, 924719236101]$	$0$	$0$	$2$	$2$	$1$	8.0.7331636232.1	$C_2 \wr S_4$	simple
4.2.ac_b_ab_d	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} - x^{3} + 3 x^{4} - 2 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 4, 19, 36, 90, 85, 235, 553, 958]$	$4$	$[4, 184, 2764, 72496, 1114924, 25301656, 192155296, 3976405600, 74203866964, 1030910016904]$	$1$	$0$	$2$	$2$	$1$	8.0.77464805625.1	$S_4\times C_2$	simple
4.2.ac_b_ab_e	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓	✓	$1 - 2 x + x^{2} - x^{3} + 4 x^{4} - 2 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$1$	$[1, 3, 4, 23, 46, 108, 120, 279, 589, 928]$	$5$	$[5, 235, 3215, 99875, 1496275, 32789785, 249738760, 4714998875, 79440698495, 998684259925]$	$2$	$0$	$2$	$2$	$1$	8.0.17364305336.1	$C_2 \wr S_4$	simple
4.2.ac_b_a_b	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} + x^{4} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 7, 19, 41, 105, 99, 259, 547, 1043]$	$5$	$[5, 205, 4460, 74825, 1356025, 31451920, 212714885, 4326755625, 73223392460, 1119779680525]$	$1$	$0$	$2$	$2$	$1$	8.0.75217789184.1	$C_2 \wr S_4$	simple
4.2.ac_b_b_ad	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} + x^{3} - 3 x^{4} + 2 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 10, 11, 26, 78, 71, 219, 469, 1158]$	$4$	$[4, 136, 4996, 41616, 900604, 20553544, 170409376, 3680269344, 63356169484, 1253022953656]$	$2$	$0$	$2$	$2$	$1$	8.0.84375725625.1	$S_4\times C_2$	simple
4.2.ac_b_b_ab	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} + x^{3} - x^{4} + 2 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 10, 19, 46, 114, 113, 267, 505, 1078]$	$6$	$[6, 216, 6714, 74736, 1613526, 35530488, 236284656, 4429303776, 67504321782, 1158892460136]$	$5$	$0$	$2$	$2$	$1$	8.0.705911761.1	$\SL(2,3)$	simple
4.2.ac_b_c_af	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} + 2 x^{3} - 5 x^{4} + 4 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 13, 11, 31, 81, 99, 243, 481, 1243]$	$5$	$[5, 145, 7580, 44225, 1097275, 21982000, 213668005, 4042916825, 64678980260, 1357832824225]$	$1$	$0$	$2$	$2$	$1$	8.0.104150601728.1	$C_2 \wr S_4$	simple
4.2.ac_b_d_aj	$4$	$\F_{2}$	$2$	✓	✓	✓	✓				$1 - 2 x + x^{2} + 3 x^{3} - 9 x^{4} + 6 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 16, 3, 16, 42, 113, 235, 457, 1078]$	$4$	$[4, 88, 7900, 26224, 633644, 11644600, 237092384, 3976869600, 61463256100, 1158999507688]$	$0$	$0$	$2$	$2$	$1$	8.0.4343469025.1	$S_4\times C_2$	simple
4.2.ac_b_d_ai	$4$	$\F_{2}$	$2$	✓	✓	✓		✓	✓	✓	$1 - 2 x + x^{2} + 3 x^{3} - 8 x^{4} + 6 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,\frac{1}{2},\frac{1}{2},1,1,1]$	$1$	$3$	$1$	$4$	$0$	$1$	$[1, 3, 16, 7, 26, 60, 120, 247, 493, 1208]$	$5$	$[5, 115, 9455, 35075, 924775, 15874945, 252280040, 4172276475, 66181794755, 1314071058325]$	$1$	$0$	$2$	$2$	$1$	8.0.40020731528.1	$C_2 \wr S_4$	simple
4.2.ac_b_d_ah	$4$	$\F_{2}$	$2$	✓	✓	✓	✓		✓	✓	$1 - 2 x + x^{2} + 3 x^{3} - 7 x^{4} + 6 x^{5} + 4 x^{6} - 16 x^{7} + 16 x^{8}$	$[0,0,0,0,1,1,1,1]$	$0$	$4$	$0$	$4$	$0$	$1$	$[1, 3, 16, 11, 36, 78, 127, 251, 493, 1238]$	$6$	$[6, 144, 10854, 45216, 1273206, 21100176, 265047936, 4186006848, 66112224138, 1351564190064]$	$3$	$0$	$2$	$2$	$1$	8.0.60097292433.1	$C_2 \wr S_4$	simple

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Elliptic curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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