LMFDB - Elliptic curves over $\Q$ of conductor 912

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Conductor	Discriminant	j-invariant	Rank

Bad$\ p$	Curves per isogeny class	Complex multiplication	Torsion

Isogeny class degree	Cyclic isogeny degree	Isogeny class size	Integral points

Analytic order of Ш	$p\ $div$\ $\|Ш\|	Regulator	Reduction	Faltings height

Galois image	Adelic level	Adelic index	Adelic genus

Nonmax$\ \ell$	$abc$ quality	Szpiro ratio

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Results (28 matches)

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	Sato-Tate	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Adelic level	Adelic index	Adelic genus	Regulator	$Ш_{\textrm{an}}$	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
912.a1	912a1	912.a	912a	$1$	$1$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{8} \cdot 3^{6} \cdot 19 $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$				$38$	$2$	$0$	$1.134807958$	$1$	$2$	$192$	$-0.016677$	$-81415168/13851$	$0.91437$	$3.52395$	$[0, -1, 0, -57, -171]$	$y^2=x^3-x^2-57x-171$	38.2.0.a.1	$[(12, 27)]$
912.b1	912g3	912.b	912g	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{12} \cdot 3^{4} \cdot 19 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.8	2B	$456$	$48$	$0$	$2.601893913$	$1$	$3$	$384$	$0.448139$	$115714886617/1539$	$0.98111$	$4.95803$	$[0, -1, 0, -1624, -24656]$	$y^2=x^3-x^2-1624x-24656$	2.3.0.a.1, 4.12.0-4.c.1.2, 24.24.0-24.z.1.16, 76.24.0.?, 456.48.0.?	$[(58, 270)]$
912.b2	912g2	912.b	912g	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{12} \cdot 3^{2} \cdot 19^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.1	2Cs	$228$	$48$	$0$	$1.300946956$	$1$	$9$	$192$	$0.101565$	$30664297/3249$	$0.90727$	$3.74967$	$[0, -1, 0, -104, -336]$	$y^2=x^3-x^2-104x-336$	2.6.0.a.1, 4.12.0-2.a.1.1, 12.24.0-12.b.1.1, 76.24.0.?, 228.48.0.?	$[(-6, 6)]$
912.b3	912g1	912.b	912g	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{12} \cdot 3 \cdot 19 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.6	2B	$456$	$48$	$0$	$0.650473478$	$1$	$7$	$96$	$-0.245009$	$389017/57$	$0.96267$	$3.10890$	$[0, -1, 0, -24, 48]$	$y^2=x^3-x^2-24x+48$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.5, 12.12.0-4.c.1.1, 24.24.0-24.z.1.12, $\ldots$	$[(2, 2)]$
912.b4	912g4	912.b	912g	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{12} \cdot 3 \cdot 19^{4} $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.7	2B	$456$	$48$	$0$	$2.601893913$	$1$	$7$	$384$	$0.448139$	$67419143/390963$	$0.97474$	$4.18797$	$[0, -1, 0, 136, -1872]$	$y^2=x^3-x^2+136x-1872$	2.3.0.a.1, 4.12.0-4.c.1.1, 6.6.0.a.1, 12.24.0-12.g.1.2, 152.24.0.?, $\ldots$	$[(18, 78)]$
912.c1	912e3	912.c	912e	$4$	$6$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{14} \cdot 3 \cdot 19^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.4, 3.4.0.1	2B, 3B	$456$	$96$	$1$	$1$	$1$	$1$	$864$	$0.798649$	$8671983378625/82308$	$1.00775$	$5.59139$	$[0, -1, 0, -6848, 220416]$	$y^2=x^3-x^2-6848x+220416$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.d.1, 12.24.0-6.a.1.9, $\ldots$	$[ ]$
912.c2	912e4	912.c	912e	$4$	$6$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{13} \cdot 3^{2} \cdot 19^{6} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.5, 3.4.0.1	2B, 3B	$456$	$96$	$1$	$1$	$1$	$1$	$1728$	$1.145224$	$-8078253774625/846825858$	$1.01015$	$5.60541$	$[0, -1, 0, -6688, 231040]$	$y^2=x^3-x^2-6688x+231040$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.a.1, 12.24.0-6.a.1.9, $\ldots$	$[ ]$
912.c3	912e1	912.c	912e	$4$	$6$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{18} \cdot 3^{3} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.4, 3.4.0.1	2B, 3B	$456$	$96$	$1$	$1$	$1$	$1$	$288$	$0.249343$	$57066625/32832$	$1.04766$	$3.84080$	$[0, -1, 0, -128, 0]$	$y^2=x^3-x^2-128x$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.d.1, 12.24.0-6.a.1.3, $\ldots$	$[ ]$
912.c4	912e2	912.c	912e	$4$	$6$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{15} \cdot 3^{6} \cdot 19^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.5, 3.4.0.1	2B, 3B	$456$	$96$	$1$	$1$	$1$	$1$	$576$	$0.595917$	$3616805375/2105352$	$1.07346$	$4.44956$	$[0, -1, 0, 512, -512]$	$y^2=x^3-x^2+512x-512$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.a.1, 12.24.0-6.a.1.3, $\ldots$	$[ ]$
912.d1	912f2	912.d	912f	$2$	$5$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{12} \cdot 3^{2} \cdot 19^{5} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$5$	5.12.0.2	5B.4.2	$380$	$48$	$1$	$0.226284916$	$1$	$4$	$2400$	$1.344555$	$-9358714467168256/22284891$	$1.06833$	$6.61609$	$[0, -1, 0, -70245, 7189389]$	$y^2=x^3-x^2-70245x+7189389$	5.12.0.a.2, 20.24.0-5.a.2.2, 38.2.0.a.1, 190.24.1.?, 380.48.1.?	$[(156, 57)]$
912.d2	912f1	912.d	912f	$2$	$5$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{12} \cdot 3^{10} \cdot 19 $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$5$	5.12.0.1	5B.4.1	$380$	$48$	$1$	$1.131424581$	$1$	$2$	$480$	$0.539836$	$841232384/1121931$	$1.00490$	$4.27459$	$[0, -1, 0, 315, 2349]$	$y^2=x^3-x^2+315x+2349$	5.12.0.a.1, 20.24.0-5.a.1.2, 38.2.0.a.1, 190.24.1.?, 380.48.1.?	$[(36, 243)]$
912.e1	912b3	912.e	912b	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{11} \cdot 3^{3} \cdot 19^{4} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.7	2B	$456$	$48$	$0$	$1$	$1$	$1$	$384$	$0.666203$	$111223479026/3518667$	$0.97692$	$4.85052$	$[0, -1, 0, -1272, -16560]$	$y^2=x^3-x^2-1272x-16560$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.3, 12.12.0-4.c.1.2, 24.24.0-24.s.1.3, $\ldots$	$[ ]$
912.e2	912b2	912.e	912b	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{10} \cdot 3^{6} \cdot 19^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.1	2Cs	$456$	$48$	$0$	$1$	$1$	$3$	$192$	$0.319629$	$768400132/263169$	$0.94504$	$4.01889$	$[0, -1, 0, -192, 720]$	$y^2=x^3-x^2-192x+720$	2.6.0.a.1, 8.12.0-2.a.1.1, 12.12.0-2.a.1.1, 24.24.0-24.b.1.3, 76.12.0.?, $\ldots$	$[ ]$
912.e3	912b1	912.e	912b	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{8} \cdot 3^{3} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.12	2B	$456$	$48$	$0$	$1$	$1$	$1$	$96$	$-0.026945$	$2211014608/513$	$0.92081$	$3.97056$	$[0, -1, 0, -172, 928]$	$y^2=x^3-x^2-172x+928$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.2, 12.12.0-4.c.1.1, 24.24.0-24.y.1.10, $\ldots$	$[ ]$
912.e4	912b4	912.e	912b	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{11} \cdot 3^{12} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.8	2B	$456$	$48$	$0$	$1$	$1$	$1$	$384$	$0.666203$	$9878111854/10097379$	$0.98646$	$4.49528$	$[0, -1, 0, 568, 4368]$	$y^2=x^3-x^2+568x+4368$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.4, 24.24.0-24.y.1.16, 76.12.0.?, $\ldots$	$[ ]$
912.f1	912i1	912.f	912i	$1$	$1$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{8} \cdot 3^{2} \cdot 19 $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$				$38$	$2$	$0$	$0.344049278$	$1$	$6$	$96$	$-0.429106$	$8192/171$	$0.94370$	$2.65763$	$[0, 1, 0, 3, -9]$	$y^2=x^3+x^2+3x-9$	38.2.0.a.1	$[(3, 6)]$
912.g1	912l1	912.g	912l	$1$	$1$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{12} \cdot 3^{2} \cdot 19 $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$				$38$	$2$	$0$	$1$	$1$	$0$	$160$	$-0.143401$	$-1404928/171$	$0.86512$	$3.32531$	$[0, 1, 0, -37, -109]$	$y^2=x^3+x^2-37x-109$	38.2.0.a.1	$[ ]$
912.h1	912h1	912.h	912h	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{14} \cdot 3^{5} \cdot 19 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.4	2B	$456$	$12$	$0$	$0.269535280$	$1$	$11$	$480$	$0.512457$	$96386901625/18468$	$0.97983$	$4.93122$	$[0, 1, 0, -1528, 22484]$	$y^2=x^3+x^2-1528x+22484$	2.3.0.a.1, 8.6.0.d.1, 114.6.0.?, 456.12.0.?	$[(20, 18)]$
912.h2	912h2	912.h	912h	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{13} \cdot 3^{10} \cdot 19^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.5	2B	$456$	$12$	$0$	$0.134767640$	$1$	$15$	$960$	$0.859030$	$-69173457625/42633378$	$0.99175$	$4.98893$	$[0, 1, 0, -1368, 27540]$	$y^2=x^3+x^2-1368x+27540$	2.3.0.a.1, 8.6.0.a.1, 228.6.0.?, 456.12.0.?	$[(-18, 216)]$
912.i1	912c1	912.i	912c	$1$	$1$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{8} \cdot 3^{2} \cdot 19^{3} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$				$38$	$2$	$0$	$1$	$1$	$0$	$192$	$0.068805$	$70575104/61731$	$0.95058$	$3.46517$	$[0, 1, 0, 55, -93]$	$y^2=x^3+x^2+55x-93$	38.2.0.a.1	$[ ]$
912.j1	912j2	912.j	912j	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{8} \cdot 3^{6} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$228$	$12$	$0$	$1$	$1$	$1$	$144$	$0.022556$	$340062928/13851$	$0.89938$	$3.69588$	$[0, 1, 0, -92, -360]$	$y^2=x^3+x^2-92x-360$	2.3.0.a.1, 12.6.0.c.1, 76.6.0.?, 228.12.0.?	$[ ]$
912.j2	912j1	912.j	912j	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{4} \cdot 3^{3} \cdot 19^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$228$	$12$	$0$	$1$	$1$	$1$	$72$	$-0.324018$	$131072/9747$	$1.13743$	$2.84701$	$[0, 1, 0, 3, -18]$	$y^2=x^3+x^2+3x-18$	2.3.0.a.1, 6.6.0.a.1, 76.6.0.?, 228.12.0.?	$[ ]$
912.k1	912k3	912.k	912k	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{17} \cdot 3^{3} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.104	2B	$456$	$48$	$0$	$1$	$1$	$1$	$5760$	$1.794586$	$74220219816682217473/16416$	$1.10905$	$7.93342$	$[0, 1, 0, -1400832, 637689780]$	$y^2=x^3+x^2-1400832x+637689780$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.m.1.7, 228.12.0.?, 456.48.0.?	$[ ]$
912.k2	912k2	912.k	912k	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{22} \cdot 3^{6} \cdot 19^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.3	2Cs	$456$	$48$	$0$	$1$	$1$	$3$	$2880$	$1.448013$	$18120364883707393/269485056$	$1.09068$	$6.71303$	$[0, 1, 0, -87552, 9941940]$	$y^2=x^3+x^2-87552x+9941940$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.b.1.1, 228.24.0.?, 456.48.0.?	$[ ]$
912.k3	912k4	912.k	912k	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{17} \cdot 3^{12} \cdot 19^{4} $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.48	2B	$456$	$48$	$0$	$1$	$1$	$3$	$5760$	$1.794586$	$-16576888679672833/2216253521952$	$1.04427$	$6.73046$	$[0, 1, 0, -84992, 10553268]$	$y^2=x^3+x^2-84992x+10553268$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.d.1.2, 456.48.0.?	$[ ]$
912.k4	912k1	912.k	912k	$4$	$4$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{32} \cdot 3^{3} \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.63	2B	$456$	$48$	$0$	$1$	$1$	$1$	$1440$	$1.101439$	$4824238966273/537919488$	$1.00823$	$5.50534$	$[0, 1, 0, -5632, 144308]$	$y^2=x^3+x^2-5632x+144308$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.m.1.3, 114.6.0.?, 228.24.0.?, $\ldots$	$[ ]$
912.l1	912d1	912.l	912d	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ 2^{10} \cdot 3 \cdot 19 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.4	2B	$456$	$12$	$0$	$1$	$1$	$1$	$160$	$-0.354923$	$470596/57$	$0.94139$	$2.93343$	$[0, 1, 0, -16, -28]$	$y^2=x^3+x^2-16x-28$	2.3.0.a.1, 8.6.0.d.1, 114.6.0.?, 456.12.0.?	$[ ]$
912.l2	912d2	912.l	912d	$2$	$2$	$ 2^{4} \cdot 3 \cdot 19 $	$ - 2^{11} \cdot 3^{2} \cdot 19^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.5	2B	$456$	$12$	$0$	$1$	$1$	$1$	$320$	$-0.008349$	$715822/3249$	$0.90786$	$3.37885$	$[0, 1, 0, 24, -108]$	$y^2=x^3+x^2+24x-108$	2.3.0.a.1, 8.6.0.a.1, 228.6.0.?, 456.12.0.?	$[ ]$

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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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