Elliptic curve search results

Results (10 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
12.1-a6	12.1-a	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	12.1	\( 2^{2} \cdot 3 \)	\( - 2^{37} \cdot 3 \)	$0.95541$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$7.858181321$	1.367933784	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[1\) , \( 0\) , \( a\) , \( 22 a - 88\) , \( -134 a + 464\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}+\left(22a-88\right){x}-134a+464$
12.1-b6	12.1-b	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	12.1	\( 2^{2} \cdot 3 \)	\( - 2^{37} \cdot 3 \)	$0.95541$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 3$	2B, 3B.1.2	$9$	\( 2 \)	$1$	$1.746262515$	1.367933784	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -4468 a - 10599\) , \( -277078 a - 657307\bigr] \)	${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4468a-10599\right){x}-277078a-657307$
288.3-e6	288.3-e	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.3	\( 2^{5} \cdot 3^{2} \)	\( - 2^{49} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$9$	\( 2^{3} \)	$1$	$0.504102566$	1.579553877	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( 137863 a - 464912\) , \( 58034539 a - 195708792\bigr] \)	${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(137863a-464912\right){x}+58034539a-195708792$
288.3-l6	288.3-l	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.3	\( 2^{5} \cdot 3^{2} \)	\( - 2^{49} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$3.098072458$	$2.268461550$	4.893572364	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -50 a - 496\) , \( 455 a + 4456\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-50a-496\right){x}+455a+4456$
288.4-e6	288.4-e	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.4	\( 2^{5} \cdot 3^{2} \)	\( - 2^{49} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{4} \)	$1$	$2.268461550$	1.579553877	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 1332 a - 4524\) , \( -54624 a + 184272\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1332a-4524\right){x}-54624a+184272$
288.4-l6	288.4-l	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.4	\( 2^{5} \cdot 3^{2} \)	\( - 2^{49} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{6} \cdot 3^{2} \)	$0.774518114$	$0.504102566$	4.893572364	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -11842 a - 28096\) , \( -1166836 a - 2768069\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11842a-28096\right){x}-1166836a-2768069$
384.5-c6	384.5-c	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.5	\( 2^{7} \cdot 3 \)	\( - 2^{55} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$9$	\( 2^{2} \)	$1$	$1.634325113$	2.560495363	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a\) , \( -a\) , \( a\) , \( -50826 a - 120572\) , \( -10353069 a - 24560392\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-50826a-120572\right){x}-10353069a-24560392$
384.5-r6	384.5-r	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.5	\( 2^{7} \cdot 3 \)	\( - 2^{55} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$6.224164637$	$1.049550007$	2.274349657	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a\) , \( 1\) , \( 0\) , \( 115 a - 521\) , \( 2035 a - 6425\bigr] \)	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(115a-521\right){x}+2035a-6425$
384.6-c6	384.6-c	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.6	\( 2^{7} \cdot 3 \)	\( - 2^{55} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{4} \cdot 3^{2} \)	$1$	$1.634325113$	2.560495363	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 266 a - 973\) , \( -4997 a + 16579\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(266a-973\right){x}-4997a+16579$
384.6-r6	384.6-r	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.6	\( 2^{7} \cdot 3 \)	\( - 2^{55} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$6.224164637$	$1.049550007$	2.274349657	\( \frac{1710723757560125}{206158430208} a + \frac{3358951907790875}{206158430208} \)	\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -25149 a - 59661\) , \( 3538879 a + 8395216\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-25149a-59661\right){x}+3538879a+8395216$

 

  *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.