LMFDB - Elliptic curves over $\Q$ of conductor 672

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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 (24 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
672.a1	672d2	672.a	672d	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{10} \cdot 7^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$1$	$1$	$1$	$960$	$0.808655$	$92100460096/20253807$	$1.03479$	$5.15554$	$[0, -1, 0, -1505, -17199]$	$y^2=x^3-x^2-1505x-17199$	2.3.0.a.1, 12.6.0.c.1, 28.6.0.a.1, 84.12.0.?	$[]$
672.a2	672d1	672.a	672d	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{6} \cdot 3^{5} \cdot 7^{6} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$1$	$1$	$1$	$480$	$0.462081$	$15926924096/28588707$	$1.04804$	$4.36126$	$[0, -1, 0, 210, -1764]$	$y^2=x^3-x^2+210x-1764$	2.3.0.a.1, 6.6.0.a.1, 28.6.0.b.1, 84.12.0.?	$[]$
672.b1	672e3	672.b	672e	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{9} \cdot 3 \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.6	2B	$168$	$48$	$0$	$1.986218322$	$1$	$3$	$128$	$-0.058633$	$2438569736/21$	$0.97593$	$4.27833$	$[0, -1, 0, -224, 1368]$	$y^2=x^3-x^2-224x+1368$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.5, 24.24.0-24.bb.1.6, 28.12.0-4.c.1.1, $\ldots$	$[(13, 22)]$
672.b2	672e2	672.b	672e	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{4} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.8	2B	$168$	$48$	$0$	$0.496554580$	$1$	$9$	$128$	$-0.058633$	$3241792/567$	$0.96732$	$3.58041$	$[0, -1, 0, -49, -95]$	$y^2=x^3-x^2-49x-95$	2.3.0.a.1, 4.12.0-4.c.1.2, 24.24.0-24.bb.1.12, 28.24.0-28.h.1.1, 168.48.0.?	$[(-3, 4)]$
672.b3	672e1	672.b	672e	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{6} \cdot 3^{2} \cdot 7^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.1	2Cs	$168$	$48$	$0$	$0.993109161$	$1$	$9$	$64$	$-0.405207$	$5088448/441$	$0.96662$	$3.01084$	$[0, -1, 0, -14, 24]$	$y^2=x^3-x^2-14x+24$	2.6.0.a.1, 4.12.0-2.a.1.1, 24.24.0-24.a.1.1, 28.24.0-28.a.1.2, 168.48.0.?	$[(4, 4)]$
672.b4	672e4	672.b	672e	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{9} \cdot 3 \cdot 7^{4} $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.7	2B	$168$	$48$	$0$	$1.986218322$	$1$	$7$	$128$	$-0.058633$	$830584/7203$	$0.97109$	$3.45704$	$[0, -1, 0, 16, 84]$	$y^2=x^3-x^2+16x+84$	2.3.0.a.1, 4.12.0-4.c.1.1, 24.24.0-24.v.1.2, 56.24.0-56.ba.1.4, 168.48.0.?	$[(1, 10)]$
672.c1	672a2	672.c	672a	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{2} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$0.307185701$	$1$	$11$	$64$	$-0.210543$	$1000000/63$	$1.02461$	$3.39975$	$[0, -1, 0, -33, 81]$	$y^2=x^3-x^2-33x+81$	2.3.0.a.1, 12.6.0.c.1, 28.6.0.a.1, 84.12.0.?	$[(5, 4)]$
672.c2	672a1	672.c	672a	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{6} \cdot 3 \cdot 7^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$0.614371403$	$1$	$5$	$32$	$-0.557117$	$8000/147$	$0.94365$	$2.54553$	$[0, -1, 0, 2, 4]$	$y^2=x^3-x^2+2x+4$	2.3.0.a.1, 6.6.0.a.1, 28.6.0.b.1, 84.12.0.?	$[(0, 2)]$
672.d1	672c3	672.d	672c	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{2} \cdot 7 $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.50	2B	$56$	$48$	$0$	$1$	$1$	$3$	$128$	$0.098947$	$1036433728/63$	$1.07156$	$4.46631$	$[0, -1, 0, -337, 2497]$	$y^2=x^3-x^2-337x+2497$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.p.1.1, 28.24.0-28.h.1.2, 56.48.0-56.br.1.6	$[]$
672.d2	672c2	672.d	672c	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{9} \cdot 3^{2} \cdot 7^{4} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.58	2B	$56$	$48$	$0$	$1$	$1$	$1$	$128$	$0.098947$	$306182024/21609$	$0.95815$	$3.95960$	$[0, -1, 0, -112, -392]$	$y^2=x^3-x^2-112x-392$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.k.1.1, 56.48.0-56.bc.1.8	$[]$
672.d3	672c1	672.d	672c	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{6} \cdot 3^{4} \cdot 7^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.1	2Cs	$56$	$48$	$0$	$1$	$1$	$3$	$64$	$-0.247627$	$19248832/3969$	$1.05560$	$3.21521$	$[0, -1, 0, -22, 40]$	$y^2=x^3-x^2-22x+40$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.a.1.2, 28.24.0-28.a.1.1, 56.48.0-56.b.1.7	$[]$
672.d4	672c4	672.d	672c	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{9} \cdot 3^{8} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.103	2B	$56$	$48$	$0$	$1$	$1$	$1$	$128$	$0.098947$	$23393656/45927$	$1.19818$	$3.69857$	$[0, -1, 0, 48, 180]$	$y^2=x^3-x^2+48x+180$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.p.1.7, 28.12.0-4.c.1.2, 56.48.0-56.bt.1.3	$[]$
672.e1	672b2	672.e	672b	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{10} \cdot 7^{3} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$0.069974632$	$1$	$23$	$960$	$0.808655$	$92100460096/20253807$	$1.03479$	$5.15554$	$[0, 1, 0, -1505, 17199]$	$y^2=x^3+x^2-1505x+17199$	2.3.0.a.1, 12.6.0.c.1, 28.6.0.a.1, 84.12.0.?	$[(7, 84)]$
672.e2	672b1	672.e	672b	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{6} \cdot 3^{5} \cdot 7^{6} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$0.139949265$	$1$	$13$	$480$	$0.462081$	$15926924096/28588707$	$1.04804$	$4.36126$	$[0, 1, 0, 210, 1764]$	$y^2=x^3+x^2+210x+1764$	2.3.0.a.1, 6.6.0.a.1, 28.6.0.b.1, 84.12.0.?	$[(0, 42)]$
672.f1	672f2	672.f	672f	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{9} \cdot 3 \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.12.0.6	2B	$168$	$48$	$0$	$3.582983643$	$1$	$3$	$128$	$-0.058633$	$2438569736/21$	$0.97593$	$4.27833$	$[0, 1, 0, -224, -1368]$	$y^2=x^3+x^2-224x-1368$	2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.5, 24.24.0-24.bb.1.14, 28.12.0-4.c.1.2, $\ldots$	$[(27, 114)]$
672.f2	672f3	672.f	672f	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{4} \cdot 7 $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.7	2B	$168$	$48$	$0$	$0.895745910$	$1$	$13$	$128$	$-0.058633$	$3241792/567$	$0.96732$	$3.58041$	$[0, 1, 0, -49, 95]$	$y^2=x^3+x^2-49x+95$	2.3.0.a.1, 4.12.0-4.c.1.1, 24.24.0-24.bb.1.4, 28.24.0-28.h.1.2, 168.48.0.?	$[(2, 3)]$
672.f3	672f1	672.f	672f	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{6} \cdot 3^{2} \cdot 7^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.1	2Cs	$168$	$48$	$0$	$1.791491821$	$1$	$7$	$64$	$-0.405207$	$5088448/441$	$0.96662$	$3.01084$	$[0, 1, 0, -14, -24]$	$y^2=x^3+x^2-14x-24$	2.6.0.a.1, 4.12.0-2.a.1.1, 24.24.0-24.a.1.4, 28.24.0-28.a.1.1, 168.48.0.?	$[(6, 12)]$
672.f4	672f4	672.f	672f	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{9} \cdot 3 \cdot 7^{4} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.12.0.8	2B	$168$	$48$	$0$	$3.582983643$	$1$	$1$	$128$	$-0.058633$	$830584/7203$	$0.97109$	$3.45704$	$[0, 1, 0, 16, -84]$	$y^2=x^3+x^2+16x-84$	2.3.0.a.1, 4.12.0-4.c.1.2, 24.24.0-24.v.1.1, 56.24.0-56.ba.1.12, 168.48.0.?	$[(21/2, 105/2)]$
672.g1	672g2	672.g	672g	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{2} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$1$	$1$	$1$	$64$	$-0.210543$	$1000000/63$	$1.02461$	$3.39975$	$[0, 1, 0, -33, -81]$	$y^2=x^3+x^2-33x-81$	2.3.0.a.1, 12.6.0.c.1, 28.6.0.a.1, 84.12.0.?	$[]$
672.g2	672g1	672.g	672g	$2$	$2$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{6} \cdot 3 \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	2.3.0.1	2B	$84$	$12$	$0$	$1$	$1$	$1$	$32$	$-0.557117$	$8000/147$	$0.94365$	$2.54553$	$[0, 1, 0, 2, -4]$	$y^2=x^3+x^2+2x-4$	2.3.0.a.1, 6.6.0.a.1, 28.6.0.b.1, 84.12.0.?	$[]$
672.h1	672h2	672.h	672h	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{12} \cdot 3^{2} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.60	2B	$56$	$48$	$0$	$1$	$1$	$1$	$128$	$0.098947$	$1036433728/63$	$1.07156$	$4.46631$	$[0, 1, 0, -337, -2497]$	$y^2=x^3+x^2-337x-2497$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.p.1.3, 28.24.0-28.h.1.1, 56.48.0-56.br.1.2	$[]$
672.h2	672h3	672.h	672h	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{9} \cdot 3^{2} \cdot 7^{4} $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.49	2B	$56$	$48$	$0$	$1$	$1$	$3$	$128$	$0.098947$	$306182024/21609$	$0.95815$	$3.95960$	$[0, 1, 0, -112, 392]$	$y^2=x^3+x^2-112x+392$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.k.1.2, 56.48.0-56.bc.1.3	$[]$
672.h3	672h1	672.h	672h	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ 2^{6} \cdot 3^{4} \cdot 7^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.2	2Cs	$56$	$48$	$0$	$1$	$1$	$3$	$64$	$-0.247627$	$19248832/3969$	$1.05560$	$3.21521$	$[0, 1, 0, -22, -40]$	$y^2=x^3+x^2-22x-40$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.a.1.1, 28.24.0-28.a.1.2, 56.48.0-56.b.1.1	$[]$
672.h4	672h4	672.h	672h	$4$	$4$	$ 2^{5} \cdot 3 \cdot 7 $	$ - 2^{9} \cdot 3^{8} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.101	2B	$56$	$48$	$0$	$1$	$1$	$1$	$128$	$0.098947$	$23393656/45927$	$1.19818$	$3.69857$	$[0, 1, 0, 48, -180]$	$y^2=x^3+x^2+48x-180$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.p.1.5, 28.12.0-4.c.1.1, 56.48.0-56.bt.1.7	$[]$

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Elliptic curves over $\Q$
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Genus 2 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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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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