Elliptic curve search results

Base field	Conductor norm	Rank*	Torsion order	CM discriminant
Base field degree/signature	Bad \(p\)	$\Q$-curves	Torsion structure	CM
Analytic order of Ш*	Cyclic isogeny degree	Isogeny class size	Reduction	Galois image
j-invariant	Regulator*	Curves per isogeny class	Isogeny class degree	Nonmax$\ \ell$

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Results (12 matches)

 

Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
1536.1-b1	1536.1-b	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	1536.1	\( 2^{9} \cdot 3 \)	\( 2^{20} \cdot 3^{4} \)	$1.93788$	$(a+1), (a)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{2} \)	$1$	$6.070636265$	1.752441741	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 4 a + 8\) , \( -12 a - 20\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a+8\right){x}-12a-20$
1536.1-k1	1536.1-k	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	1536.1	\( 2^{9} \cdot 3 \)	\( 2^{20} \cdot 3^{4} \)	$1.93788$	$(a+1), (a)$	$0$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{4} \)	$1$	$6.070636265$	1.752441741	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a + 8\) , \( -12 a + 20\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+8\right){x}-12a+20$
1536.1-n1	1536.1-n	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	1536.1	\( 2^{9} \cdot 3 \)	\( 2^{20} \cdot 3^{4} \)	$1.93788$	$(a+1), (a)$	$0$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{4} \)	$1$	$6.070636265$	1.752441741	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a + 8\) , \( 12 a + 20\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+8\right){x}+12a+20$
1536.1-w1	1536.1-w	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	1536.1	\( 2^{9} \cdot 3 \)	\( 2^{20} \cdot 3^{4} \)	$1.93788$	$(a+1), (a)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{2} \)	$1$	$6.070636265$	1.752441741	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a + 8\) , \( 12 a - 20\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a+8\right){x}+12a-20$
3072.1-d1	3072.1-d	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	3072.1	\( 2^{10} \cdot 3 \)	\( 2^{14} \cdot 3^{4} \)	$2.30454$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$0.603762616$	$6.405092923$	4.465406728	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a + 13\) , \( 23 a - 39\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a+13\right){x}+23a-39$
3072.1-bb1	3072.1-bb	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	3072.1	\( 2^{10} \cdot 3 \)	\( 2^{14} \cdot 3^{4} \)	$2.30454$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2 \)	$1.696743987$	$6.405092923$	3.137264466	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -94 a + 163\) , \( 1183 a - 2049\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-94a+163\right){x}+1183a-2049$
3072.1-bn1	3072.1-bn	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	3072.1	\( 2^{10} \cdot 3 \)	\( 2^{14} \cdot 3^{4} \)	$2.30454$	$(a+1), (a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{2} \)	$0.298069290$	$11.50728806$	3.960587271	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a + 13\) , \( -23 a + 39\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+13\right){x}-23a+39$
3072.1-bu1	3072.1-bu	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	3072.1	\( 2^{10} \cdot 3 \)	\( 2^{14} \cdot 3^{4} \)	$2.30454$	$(a+1), (a)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{2} \)	$1$	$11.50728806$	3.321867930	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -94 a + 163\) , \( -1183 a + 2049\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-94a+163\right){x}-1183a+2049$
4608.1-g1	4608.1-g	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	4608.1	\( 2^{9} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{10} \)	$2.55039$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{4} \)	$0.373870495$	$4.697830679$	4.056186517	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -10 a + 21\) , \( -49 a + 86\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(-10a+21\right){x}-49a+86$
4608.1-j1	4608.1-j	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	4608.1	\( 2^{9} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{10} \)	$2.55039$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{4} \)	$0.373870495$	$4.697830679$	4.056186517	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a + 21\) , \( 49 a + 86\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(10a+21\right){x}+49a+86$
4608.1-r1	4608.1-r	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	4608.1	\( 2^{9} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{10} \)	$2.55039$	$(a+1), (a)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{3} \)	$1$	$2.614868236$	1.509694880	\( \frac{4000}{9} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -10 a + 21\) , \( 49 a - 86\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(-10a+21\right){x}+49a-86$
4608.1-bd1	4608.1-bd	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	4608.1	\( 2^{9} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{10} \)	$2.55039$	$(a+1), (a)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{3} \)	$1$	$2.614868236$	1.509694880	\( \frac{4000}{9} \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a + 21\) , \( -49 a - 86\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(10a+21\right){x}-49a-86$

 

  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.