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-a1	12.1-a	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	12.1	\( 2^{2} \cdot 3 \)	\( - 2^{13} \cdot 3 \)	$0.95541$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 3$	2B, 3B.1.1	$1$	\( 2 \cdot 3^{2} \)	$1$	$15.71636264$	1.367933784	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 43110 a - 145374\) , \( -8233861 a + 27766893\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(43110a-145374\right){x}-8233861a+27766893$
12.1-b1	12.1-b	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	12.1	\( 2^{2} \cdot 3 \)	\( - 2^{13} \cdot 3 \)	$0.95541$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 3$	2B, 3B.1.2	$9$	\( 2^{2} \)	$1$	$0.873131257$	1.367933784	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -23 a - 171\) , \( -182 a - 930\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-23a-171\right){x}-182a-930$
288.3-e1	288.3-e	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.3	\( 2^{5} \cdot 3^{2} \)	\( - 2^{25} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$9$	\( 2^{4} \)	$1$	$0.252051283$	1.579553877	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -2277 a - 5712\) , \( -106971 a - 255912\bigr] \)	${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-2277a-5712\right){x}-106971a-255912$
288.3-l1	288.3-l	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.3	\( 2^{5} \cdot 3^{2} \)	\( - 2^{25} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{6} \)	$0.774518114$	$4.536923101$	4.893572364	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 114215 a - 385176\) , \( -35594295 a + 120033976\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(114215a-385176\right){x}-35594295a+120033976$
288.4-e1	288.4-e	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.4	\( 2^{5} \cdot 3^{2} \)	\( - 2^{25} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$1$	$4.536923101$	1.579553877	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -240238 a - 569914\) , \( 110766716 a + 262769812\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-240238a-569914\right){x}+110766716a+262769812$
288.4-l1	288.4-l	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	288.4	\( 2^{5} \cdot 3^{2} \)	\( - 2^{25} \cdot 3^{7} \)	$2.11468$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{3} \cdot 3^{2} \)	$3.098072458$	$0.252051283$	4.893572364	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 993 a - 4011\) , \( 33564 a - 119829\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(993a-4011\right){x}+33564a-119829$
384.5-c1	384.5-c	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.5	\( 2^{7} \cdot 3 \)	\( - 2^{31} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$9$	\( 2^{4} \)	$1$	$1.634325113$	2.560495363	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -373 a - 1545\) , \( -11804 a - 22017\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-373a-1545\right){x}-11804a-22017$
384.5-r1	384.5-r	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.5	\( 2^{7} \cdot 3 \)	\( - 2^{31} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$6.224164637$	$1.049550007$	2.274349657	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a\) , \( a - 1\) , \( a\) , \( 242607 a - 818137\) , \( 110078707 a - 371216367\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(242607a-818137\right){x}+110078707a-371216367$
384.6-c1	384.6-c	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.6	\( 2^{7} \cdot 3 \)	\( - 2^{31} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$1.634325113$	2.560495363	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 490249 a - 1653263\) , \( -316753432 a + 1068181679\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(490249a-1653263\right){x}-316753432a+1068181679$
384.6-r1	384.6-r	$12$	$36$	\(\Q(\sqrt{33}) \)	$2$	$[2, 0]$	384.6	\( 2^{7} \cdot 3 \)	\( - 2^{31} \cdot 3 \)	$2.27237$	$(-a-2), (-a+3), (-2a+7)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$6.224164637$	$1.049550007$	2.274349657	\( -\frac{74785175353186375}{1536} a + \frac{252196650035225375}{1536} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -10 a - 1351\) , \( 6890 a - 4075\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a-1351\right){x}+6890a-4075$

 

  *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.