LMFDB - Elliptic curves over $\Q$ of conductor 192

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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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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
192.a1	192a3	192.a	192a	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{15} \cdot 3 $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.53	2B	$24$	$48$	$0$	$1.351603734$	$1$	$9$	$32$	$-0.079647$	$7301384/3$	$1.03749$	$4.98351$	$[0, -1, 0, -129, 609]$	$y^2=x^3-x^2-129x+609$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.m.1.1, 24.48.0-24.bi.1.2	$[(8, 5)]$
192.a2	192a2	192.a	192a	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{12} \cdot 3^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.4	2Cs	$24$	$48$	$0$	$0.675801867$	$1$	$13$	$16$	$-0.426220$	$21952/9$	$1.09175$	$3.48348$	$[0, -1, 0, -9, 9]$	$y^2=x^3-x^2-9x+9$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.b.1.2, 12.24.0-12.a.1.3, 24.48.0-24.f.1.8	$[(5, 8)]$
192.a3	192a1	192.a	192a	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{6} \cdot 3 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.102	2B	$24$	$48$	$0$	$1.351603734$	$1$	$3$	$8$	$-0.772794$	$140608/3$	$1.02705$	$3.04567$	$[0, -1, 0, -4, -2]$	$y^2=x^3-x^2-4x-2$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.m.1.5, 12.12.0.h.1, 24.48.0-24.bk.1.3	$[(3, 2)]$
192.a4	192a4	192.a	192a	$4$	$4$	$ 2^{6} \cdot 3 $	$ - 2^{15} \cdot 3^{4} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.59	2B	$24$	$48$	$0$	$0.337900933$	$1$	$11$	$32$	$-0.079647$	$97336/81$	$1.05989$	$4.16227$	$[0, -1, 0, 31, 33]$	$y^2=x^3-x^2+31x+33$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.d.1.1, 24.48.0-24.n.1.2	$[(1, 8)]$
192.b1	192d5	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{17} \cdot 3^{2} $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.113	2B	$48$	$192$	$1$	$1$	$1$	$3$	$64$	$0.394369$	$3065617154/9$	$1.21059$	$6.39601$	$[0, -1, 0, -1537, 23713]$	$y^2=x^3-x^2-1537x+23713$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.48.0-8.r.1.5, 16.96.0-16.l.1.5, 24.96.0-24.bf.1.1, $\ldots$	$[]$
192.b2	192d3	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{16} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.280	2B	$48$	$192$	$1$	$1$	$1$	$1$	$32$	$0.047795$	$28756228/3$	$1.05617$	$5.37608$	$[0, -1, 0, -257, -1503]$	$y^2=x^3-x^2-257x-1503$	2.3.0.a.1, 4.6.0.c.1, 8.48.0-8.bb.2.2, 12.12.0.h.1, 16.96.0-16.bb.2.2, $\ldots$	$[]$
192.b3	192d4	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{16} \cdot 3^{4} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.15	2Cs	$24$	$192$	$1$	$1$	$1$	$3$	$32$	$0.047795$	$1556068/81$	$1.03212$	$4.82131$	$[0, -1, 0, -97, 385]$	$y^2=x^3-x^2-97x+385$	2.6.0.a.1, 4.24.0-4.b.1.2, 8.96.0-8.e.2.6, 24.192.1-24.bl.2.7	$[]$
192.b4	192d2	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{14} \cdot 3^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.144	2Cs	$24$	$192$	$1$	$1$	$1$	$3$	$16$	$-0.298779$	$35152/9$	$0.97255$	$3.83671$	$[0, -1, 0, -17, -15]$	$y^2=x^3-x^2-17x-15$	2.6.0.a.1, 4.12.0.b.1, 8.96.0-8.h.1.5, 12.24.0.c.1, 24.192.1-24.bu.1.5	$[]$
192.b5	192d1	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ - 2^{10} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.300	2B	$48$	$192$	$1$	$1$	$1$	$1$	$8$	$-0.645352$	$2048/3$	$1.17572$	$2.84968$	$[0, -1, 0, 3, -3]$	$y^2=x^3-x^2+3x-3$	2.3.0.a.1, 4.6.0.c.1, 6.6.0.a.1, 8.48.0-8.ba.1.6, 12.12.0.g.1, $\ldots$	$[]$
192.b6	192d6	192.b	192d	$6$	$8$	$ 2^{6} \cdot 3 $	$ - 2^{17} \cdot 3^{8} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.170	2B	$48$	$192$	$1$	$1$	$1$	$1$	$64$	$0.394369$	$207646/6561$	$1.15980$	$5.32737$	$[0, -1, 0, 63, 1377]$	$y^2=x^3-x^2+63x+1377$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.96.0-8.m.1.3, 48.192.1-48.w.2.5	$[]$
192.c1	192b3	192.c	192b	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{15} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.63	2B	$24$	$48$	$0$	$1$	$1$	$1$	$32$	$-0.079647$	$7301384/3$	$1.03749$	$4.98351$	$[0, 1, 0, -129, -609]$	$y^2=x^3+x^2-129x-609$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.m.1.3, 24.48.0-24.bi.1.1	$[]$
192.c2	192b2	192.c	192b	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{12} \cdot 3^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.3	2Cs	$24$	$48$	$0$	$1$	$1$	$3$	$16$	$-0.426220$	$21952/9$	$1.09175$	$3.48348$	$[0, 1, 0, -9, -9]$	$y^2=x^3+x^2-9x-9$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.b.1.1, 12.24.0-12.a.1.3, 24.48.0-24.f.1.5	$[]$
192.c3	192b1	192.c	192b	$4$	$4$	$ 2^{6} \cdot 3 $	$ 2^{6} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.104	2B	$24$	$48$	$0$	$1$	$1$	$1$	$8$	$-0.772794$	$140608/3$	$1.02705$	$3.04567$	$[0, 1, 0, -4, 2]$	$y^2=x^3+x^2-4x+2$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.m.1.7, 12.12.0.h.1, 24.48.0-24.bk.1.2	$[]$
192.c4	192b4	192.c	192b	$4$	$4$	$ 2^{6} \cdot 3 $	$ - 2^{15} \cdot 3^{4} $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.48	2B	$24$	$48$	$0$	$1$	$1$	$3$	$32$	$-0.079647$	$97336/81$	$1.05989$	$4.16227$	$[0, 1, 0, 31, -33]$	$y^2=x^3+x^2+31x-33$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.d.1.2, 24.48.0-24.n.1.4	$[]$
192.d1	192c5	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{17} \cdot 3^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.155	2B	$48$	$192$	$1$	$1$	$1$	$1$	$64$	$0.394369$	$3065617154/9$	$1.21059$	$6.39601$	$[0, 1, 0, -1537, -23713]$	$y^2=x^3+x^2-1537x-23713$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.48.0-8.r.1.1, 16.96.0-16.l.1.1, 24.96.0-24.bf.1.5, $\ldots$	$[]$
192.d2	192c4	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{16} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.220	2B	$48$	$192$	$1$	$1$	$1$	$1$	$32$	$0.047795$	$28756228/3$	$1.05617$	$5.37608$	$[0, 1, 0, -257, 1503]$	$y^2=x^3+x^2-257x+1503$	2.3.0.a.1, 4.6.0.c.1, 8.48.0-8.bb.2.1, 12.12.0.h.1, 16.96.0-16.bb.2.1, $\ldots$	$[]$
192.d3	192c3	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{16} \cdot 3^{4} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.9	2Cs	$24$	$192$	$1$	$1$	$1$	$3$	$32$	$0.047795$	$1556068/81$	$1.03212$	$4.82131$	$[0, 1, 0, -97, -385]$	$y^2=x^3+x^2-97x-385$	2.6.0.a.1, 4.24.0-4.b.1.2, 8.96.0-8.e.2.5, 24.192.1-24.bl.2.5	$[]$
192.d4	192c2	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ 2^{14} \cdot 3^{2} $	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.124	2Cs	$24$	$192$	$1$	$1$	$1$	$3$	$16$	$-0.298779$	$35152/9$	$0.97255$	$3.83671$	$[0, 1, 0, -17, 15]$	$y^2=x^3+x^2-17x+15$	2.6.0.a.1, 4.12.0.b.1, 8.96.0-8.h.1.1, 12.24.0.c.1, 24.192.1-24.bu.1.3	$[]$
192.d5	192c1	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ - 2^{10} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.96.0.263	2B	$48$	$192$	$1$	$1$	$1$	$1$	$8$	$-0.645352$	$2048/3$	$1.17572$	$2.84968$	$[0, 1, 0, 3, 3]$	$y^2=x^3+x^2+3x+3$	2.3.0.a.1, 4.6.0.c.1, 6.6.0.a.1, 8.48.0-8.ba.1.5, 12.12.0.g.1, $\ldots$	$[]$
192.d6	192c6	192.d	192c	$6$	$8$	$ 2^{6} \cdot 3 $	$ - 2^{17} \cdot 3^{8} $	$0$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.96.0.159	2B	$48$	$192$	$1$	$1$	$1$	$3$	$64$	$0.394369$	$207646/6561$	$1.15980$	$5.32737$	$[0, 1, 0, 63, -1377]$	$y^2=x^3+x^2+63x-1377$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.96.0-8.m.1.1, 48.192.1-48.w.2.7	$[]$

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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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