Elliptic curves over $\Q$ of conductor 21

Conductor	j-invariant	Rank	Torsion	Complex multiplication
Bad$\ p$	Discriminant	Regulator	Analytic order of Ш	Galois image
Isogeny class size	Isogeny class degree	Cyclic isogeny degree	$p\ $div$\ $|Ш|	Nonmax$\ \ell$
Curves per isogeny class	Reduction	Integral points	Faltings height

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Results (6 matches)

 

Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Semistable	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	Weierstrass coefficients	Weierstrass equation
21.a1	21a5	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3 \cdot 7^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.48.0.48	2B	$1$	$1$		$0$	$4$	$0.074205$	$53297461115137/147$	$[1, 0, 0, -784, -8515]$	\(y^2+xy=x^3-784x-8515\)
21.a2	21a2	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{4} \)	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.34	2Cs	$1$	$1$		$2$	$2$	$-0.272368$	$13027640977/21609$	$[1, 0, 0, -49, -136]$	\(y^2+xy=x^3-49x-136\)
21.a3	21a3	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{8} \cdot 7 \)	$0$	$\Z/8\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.159	2B	$1$	$1$		$6$	$2$	$-0.272368$	$6570725617/45927$	$[1, 0, 0, -39, 90]$	\(y^2+xy=x^3-39x+90\)
21.a4	21a6	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( - 3 \cdot 7^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.177	2B	$1$	$1$		$0$	$4$	$0.074205$	$-4354703137/17294403$	$[1, 0, 0, -34, -217]$	\(y^2+xy=x^3-34x-217\)
21.a5	21a1	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{2} \)	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.24	2Cs	$1$	$1$		$6$	$1$	$-0.618942$	$7189057/3969$	$[1, 0, 0, -4, -1]$	\(y^2+xy=x^3-4x-1\)
21.a6	21a4	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( - 3^{2} \cdot 7 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.48.0.27	2B	$1$	$1$		$3$	$2$	$-0.965515$	$103823/63$	$[1, 0, 0, 1, 0]$	\(y^2+xy=x^3+x\)