LMFDB - Elliptic curves over $\Q$ of conductor 448

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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	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
448.a1	448f6	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{27} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$1152$	$1.452822$	$2251439055699625/25088$	$1.06489$	$7.83433$	$[0, 1, 0, -174753, 28059871]$	$y^2=x^3+x^2-174753x+28059871$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.b.1, 9.12.0.a.1, $\ldots$	$[]$
448.a2	448f5	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{36} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$576$	$1.106249$	$-548347731625/1835008$	$1.02933$	$6.47239$	$[0, 1, 0, -10913, 436447]$	$y^2=x^3+x^2-10913x+436447$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.c.1, 9.12.0.a.1, $\ldots$	$[]$
448.a3	448f4	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{21} \cdot 7^{6} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 3.12.0.1	2B, 3Cs	$504$	$864$	$21$	$1$	$1$		$1$	$384$	$0.903516$	$4956477625/941192$	$1.00821$	$5.70054$	$[0, 1, 0, -2273, 33439]$	$y^2=x^3+x^2-2273x+33439$	2.3.0.a.1, 3.12.0.a.1, 6.36.0.a.1, 8.6.0.b.1, 12.72.0-6.a.1.4, $\ldots$	$[]$
448.a4	448f2	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{19} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$128$	$0.354209$	$128787625/98$	$0.96763$	$5.10260$	$[0, 1, 0, -673, -6945]$	$y^2=x^3+x^2-673x-6945$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.b.1, 9.12.0.a.1, $\ldots$	$[]$
448.a5	448f1	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{20} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$64$	$0.007636$	$-15625/28$	$1.01712$	$3.85655$	$[0, 1, 0, -33, -161]$	$y^2=x^3+x^2-33x-161$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.c.1, 9.12.0.a.1, $\ldots$	$[]$
448.a6	448f3	448.a	448f	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{24} \cdot 7^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 3.12.0.1	2B, 3Cs	$504$	$864$	$21$	$1$	$1$		$1$	$192$	$0.556942$	$9938375/21952$	$0.98695$	$4.85261$	$[0, 1, 0, 287, 3231]$	$y^2=x^3+x^2+287x+3231$	2.3.0.a.1, 3.12.0.a.1, 6.36.0.a.1, 8.6.0.c.1, 12.72.0-6.a.1.2, $\ldots$	$[]$
448.b1	448g2	448.b	448g	$2$	$2$	$ 2^{6} \cdot 7 $	$ 2^{15} \cdot 7^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.6	2B	$56$	$12$	$0$	$0.306002722$	$1$		$9$	$64$	$-0.110627$	$125000/49$	$1.00487$	$3.62556$	$[0, 1, 0, -33, 31]$	$y^2=x^3+x^2-33x+31$	2.3.0.a.1, 8.6.0.b.1, 28.6.0.c.1, 56.12.0.k.1	$[(-1, 8)]$
448.b2	448g1	448.b	448g	$2$	$2$	$ 2^{6} \cdot 7 $	$ - 2^{12} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.1	2B	$56$	$12$	$0$	$0.612005445$	$1$		$7$	$32$	$-0.457201$	$8000/7$	$0.98030$	$2.83465$	$[0, 1, 0, 7, 7]$	$y^2=x^3+x^2+7x+7$	2.3.0.a.1, 8.6.0.c.1, 14.6.0.b.1, 56.12.0.n.1	$[(1, 4)]$
448.c1	448h2	448.c	448h	$2$	$2$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.6	2B	$56$	$12$	$0$	$1$	$1$		$1$	$128$	$0.111905$	$3543122/49$	$1.08036$	$4.40048$	$[0, 1, 0, -161, -833]$	$y^2=x^3+x^2-161x-833$	2.3.0.a.1, 8.6.0.b.1, 28.6.0.c.1, 56.12.0.k.1	$[]$
448.c2	448h1	448.c	448h	$2$	$2$	$ 2^{6} \cdot 7 $	$ - 2^{16} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.1	2B	$56$	$12$	$0$	$1$	$1$		$1$	$64$	$-0.234669$	$-4/7$	$1.03482$	$3.35659$	$[0, 1, 0, -1, -33]$	$y^2=x^3+x^2-x-33$	2.3.0.a.1, 8.6.0.c.1, 14.6.0.b.1, 56.12.0.n.1	$[]$
448.d1	448a4	448.d	448a	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7 $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.50	2B	$56$	$48$	$0$	$2.786954333$	$1$		$7$	$128$	$0.344541$	$1443468546/7$	$1.04654$	$5.38492$	$[0, 0, 0, -1196, 15920]$	$y^2=x^3-1196x+15920$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.p.1.1, 56.48.0-56.bp.1.5	$[(29, 75)]$
448.d2	448a3	448.d	448a	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7^{4} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.58	2B	$56$	$48$	$0$	$0.696738583$	$1$		$7$	$128$	$0.344541$	$11090466/2401$	$1.11706$	$4.58740$	$[0, 0, 0, -236, -1104]$	$y^2=x^3-236x-1104$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.k.1.1, 56.48.0-56.v.1.1	$[(-10, 16)]$
448.d3	448a2	448.d	448a	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{16} \cdot 7^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.1	2Cs	$56$	$48$	$0$	$1.393477166$	$1$		$9$	$64$	$-0.002033$	$740772/49$	$1.06534$	$4.03057$	$[0, 0, 0, -76, 240]$	$y^2=x^3-76x+240$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.a.1.2, 28.24.0-28.b.1.3, 56.48.0-56.d.1.7	$[(8, 12)]$
448.d4	448a1	448.d	448a	$4$	$4$	$ 2^{6} \cdot 7 $	$ - 2^{14} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.103	2B	$56$	$48$	$0$	$0.696738583$	$1$		$7$	$32$	$-0.348607$	$432/7$	$0.89152$	$3.12350$	$[0, 0, 0, 4, 16]$	$y^2=x^3+4x+16$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.p.1.7, 14.6.0.b.1, 28.12.0.g.1, $\ldots$	$[(0, 4)]$
448.e1	448b3	448.e	448b	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.60	2B	$56$	$48$	$0$	$4.276434307$	$1$		$3$	$128$	$0.344541$	$1443468546/7$	$1.04654$	$5.38492$	$[0, 0, 0, -1196, -15920]$	$y^2=x^3-1196x-15920$	2.3.0.a.1, 4.12.0-4.c.1.2, 8.24.0-8.p.1.3, 56.48.0-56.bp.1.1	$[(124, 1320)]$
448.e2	448b4	448.e	448b	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7^{4} $	$1$	$\Z/4\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.49	2B	$56$	$48$	$0$	$1.069108576$	$1$		$9$	$128$	$0.344541$	$11090466/2401$	$1.11706$	$4.58740$	$[0, 0, 0, -236, 1104]$	$y^2=x^3-236x+1104$	2.3.0.a.1, 4.12.0-4.c.1.1, 8.24.0-8.k.1.2, 56.48.0-56.v.1.6	$[(5, 7)]$
448.e3	448b2	448.e	448b	$4$	$4$	$ 2^{6} \cdot 7 $	$ 2^{16} \cdot 7^{2} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.2	2Cs	$56$	$48$	$0$	$2.138217153$	$1$		$7$	$64$	$-0.002033$	$740772/49$	$1.06534$	$4.03057$	$[0, 0, 0, -76, -240]$	$y^2=x^3-76x-240$	2.6.0.a.1, 4.12.0-2.a.1.1, 8.24.0-8.a.1.1, 28.24.0-28.b.1.3, 56.48.0-56.d.1.5	$[(12, 24)]$
448.e4	448b1	448.e	448b	$4$	$4$	$ 2^{6} \cdot 7 $	$ - 2^{14} \cdot 7 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.101	2B	$56$	$48$	$0$	$1.069108576$	$1$		$5$	$32$	$-0.348607$	$432/7$	$0.89152$	$3.12350$	$[0, 0, 0, 4, -16]$	$y^2=x^3+4x-16$	2.3.0.a.1, 4.6.0.c.1, 8.24.0-8.p.1.5, 14.6.0.b.1, 28.12.0.g.1, $\ldots$	$[(4, 8)]$
448.f1	448d2	448.f	448d	$2$	$2$	$ 2^{6} \cdot 7 $	$ 2^{15} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.6	2B	$56$	$12$	$0$	$1$	$1$		$1$	$64$	$-0.110627$	$125000/49$	$1.00487$	$3.62556$	$[0, -1, 0, -33, -31]$	$y^2=x^3-x^2-33x-31$	2.3.0.a.1, 8.6.0.b.1, 28.6.0.c.1, 56.12.0.k.1	$[]$
448.f2	448d1	448.f	448d	$2$	$2$	$ 2^{6} \cdot 7 $	$ - 2^{12} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.1	2B	$56$	$12$	$0$	$1$	$1$		$1$	$32$	$-0.457201$	$8000/7$	$0.98030$	$2.83465$	$[0, -1, 0, 7, -7]$	$y^2=x^3-x^2+7x-7$	2.3.0.a.1, 8.6.0.c.1, 14.6.0.b.1, 56.12.0.n.1	$[]$
448.g1	448c6	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{27} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$9$	$3$	$1$	$1152$	$1.452822$	$2251439055699625/25088$	$1.06489$	$7.83433$	$[0, -1, 0, -174753, -28059871]$	$y^2=x^3-x^2-174753x-28059871$	2.3.0.a.1, 3.4.0.a.1, 6.24.0-6.a.1.3, 8.6.0.b.1, 9.12.0.a.1, $\ldots$	$[]$
448.g2	448c5	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{36} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$9$	$3$	$1$	$576$	$1.106249$	$-548347731625/1835008$	$1.02933$	$6.47239$	$[0, -1, 0, -10913, -436447]$	$y^2=x^3-x^2-10913x-436447$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.c.1, 9.12.0.a.1, $\ldots$	$[]$
448.g3	448c4	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{21} \cdot 7^{6} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 3.12.0.1	2B, 3Cs	$504$	$864$	$21$	$1$	$1$		$1$	$384$	$0.903516$	$4956477625/941192$	$1.00821$	$5.70054$	$[0, -1, 0, -2273, -33439]$	$y^2=x^3-x^2-2273x-33439$	2.3.0.a.1, 3.12.0.a.1, 6.72.0-6.a.1.2, 8.6.0.b.1, 24.144.1-24.h.1.2, $\ldots$	$[]$
448.g4	448c2	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ 2^{19} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.6, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$128$	$0.354209$	$128787625/98$	$0.96763$	$5.10260$	$[0, -1, 0, -673, 6945]$	$y^2=x^3-x^2-673x+6945$	2.3.0.a.1, 3.4.0.a.1, 6.24.0-6.a.1.1, 8.6.0.b.1, 9.12.0.a.1, $\ldots$	$[]$
448.g5	448c1	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{20} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 9.12.0.1	2B, 3B	$504$	$864$	$21$	$1$	$1$		$1$	$64$	$0.007636$	$-15625/28$	$1.01712$	$3.85655$	$[0, -1, 0, -33, 161]$	$y^2=x^3-x^2-33x+161$	2.3.0.a.1, 3.4.0.a.1, 6.12.0.a.1, 8.6.0.c.1, 9.12.0.a.1, $\ldots$	$[]$
448.g6	448c3	448.g	448c	$6$	$18$	$ 2^{6} \cdot 7 $	$ - 2^{24} \cdot 7^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	8.6.0.1, 3.12.0.1	2B, 3Cs	$504$	$864$	$21$	$1$	$1$		$1$	$192$	$0.556942$	$9938375/21952$	$0.98695$	$4.85261$	$[0, -1, 0, 287, -3231]$	$y^2=x^3-x^2+287x-3231$	2.3.0.a.1, 3.12.0.a.1, 6.36.0.a.1, 8.6.0.c.1, 12.72.0-6.a.1.3, $\ldots$	$[]$
448.h1	448e2	448.h	448e	$2$	$2$	$ 2^{6} \cdot 7 $	$ 2^{17} \cdot 7^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.6	2B	$56$	$12$	$0$	$1$	$1$		$1$	$128$	$0.111905$	$3543122/49$	$1.08036$	$4.40048$	$[0, -1, 0, -161, 833]$	$y^2=x^3-x^2-161x+833$	2.3.0.a.1, 8.6.0.b.1, 28.6.0.c.1, 56.12.0.k.1	$[]$
448.h2	448e1	448.h	448e	$2$	$2$	$ 2^{6} \cdot 7 $	$ - 2^{16} \cdot 7 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.6.0.1	2B	$56$	$12$	$0$	$1$	$1$		$1$	$64$	$-0.234669$	$-4/7$	$1.03482$	$3.35659$	$[0, -1, 0, -1, 33]$	$y^2=x^3-x^2-x+33$	2.3.0.a.1, 8.6.0.c.1, 14.6.0.b.1, 56.12.0.n.1	$[]$

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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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