Elliptic curve search results

Results (17 matches)

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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
81.1-CMa2	81.1-CMa	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	81.1	\( 3^{4} \)	\( 3^{10} \)	$0.46432$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓	✓		✓	$3$	3B.1.1[2]	$1$	\( 1 \)	$1$	$2.702876088$	0.346779163	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \)	${y}^2+{y}={x}^{3}-30{x}+63$
3969.1-CMb2	3969.1-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	3969.1	\( 3^{4} \cdot 7^{2} \)	\( 3^{4} \cdot 7^{6} \)	$1.22848$	$(-2a+1), (-3a+1)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$7$	7Cs.2.1	$1$	\( 1 \)	$1$	$1.769447752$	2.043182272	\( -12288000 \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -79 a + 50\) , \( -177 a + 303\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-79a+50\right){x}-177a+303$
3969.3-CMb2	3969.3-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	3969.3	\( 3^{4} \cdot 7^{2} \)	\( 3^{4} \cdot 7^{6} \)	$1.22848$	$(-2a+1), (3a-2)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$7$	7Cs.2.1	$1$	\( 1 \)	$1$	$1.769447752$	2.043182272	\( -12288000 \)	\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 81 a - 30\) , \( 97 a + 156\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(81a-30\right){x}+97a+156$
13689.1-CMb2	13689.1-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	13689.1	\( 3^{4} \cdot 13^{2} \)	\( 3^{10} \cdot 13^{6} \)	$1.67414$	$(-2a+1), (-4a+1)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$1$	\( 3 \)	$1$	$0.749642948$	2.596839347	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -210 a - 240\) , \( 2277 a + 1075\bigr] \)	${y}^2+{y}={x}^{3}+\left(-210a-240\right){x}+2277a+1075$
13689.3-CMb2	13689.3-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	13689.3	\( 3^{4} \cdot 13^{2} \)	\( 3^{10} \cdot 13^{6} \)	$1.67414$	$(-2a+1), (4a-3)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$1$	\( 3 \)	$1$	$0.749642948$	2.596839347	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 210 a - 450\) , \( -2277 a + 3352\bigr] \)	${y}^2+{y}={x}^{3}+\left(210a-450\right){x}-2277a+3352$
20736.1-CMc2	20736.1-CMc	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	20736.1	\( 2^{8} \cdot 3^{4} \)	\( 2^{24} \cdot 3^{4} \)	$1.85729$	$(-2a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$1$	\( 1 \)	$1$	$1.170379677$	1.351438044	\( -12288000 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 161 a - 160\) , \( 846 a - 343\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(161a-160\right){x}+846a-343$
20736.1-CMb2	20736.1-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	20736.1	\( 2^{8} \cdot 3^{4} \)	\( 2^{24} \cdot 3^{4} \)	$1.85729$	$(-2a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$1$	\( 1 \)	$1$	$1.170379677$	1.351438044	\( -12288000 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -159 a\) , \( -1006 a + 503\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}-159a{x}-1006a+503$
20736.1-CMa2	20736.1-CMa	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	20736.1	\( 2^{8} \cdot 3^{4} \)	\( 2^{24} \cdot 3^{10} \)	$1.85729$	$(-2a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓	✓		✓			$1$	\( 3 \)	$1$	$0.675719022$	2.340759355	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -480 a + 480\) , \( -4048\bigr] \)	${y}^2={x}^{3}+\left(-480a+480\right){x}-4048$
29241.1-CMd2	29241.1-CMd	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	29241.1	\( 3^{4} \cdot 19^{2} \)	\( 3^{4} \cdot 19^{6} \)	$2.02394$	$(-2a+1), (-5a+3)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$19$	19Cs.4.1	$1$	\( 1 \)	$1$	$1.074014050$	1.240164602	\( -12288000 \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -49 a - 160\) , \( -480 a - 789\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-49a-160\right){x}-480a-789$
29241.3-CMd2	29241.3-CMd	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	29241.3	\( 3^{4} \cdot 19^{2} \)	\( 3^{4} \cdot 19^{6} \)	$2.02394$	$(-2a+1), (-5a+2)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$19$	19Cs.4.1	$1$	\( 1 \)	$1$	$1.074014050$	1.240164602	\( -12288000 \)	\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 51 a - 210\) , \( 430 a - 1059\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(51a-210\right){x}+430a-1059$
50625.1-CMb2	50625.1-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	50625.1	\( 3^{4} \cdot 5^{4} \)	\( 3^{10} \cdot 5^{12} \)	$2.32162$	$(-2a+1), (5)$	$2$	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓	✓		✓			$1$	\( 2^{2} \cdot 3 \)	$0.123378097$	$0.540575217$	3.696619976	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -750\) , \( 7906\bigr] \)	${y}^2+{y}={x}^{3}-750{x}+7906$
77841.1-CMc2	77841.1-CMc	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	77841.1	\( 3^{4} \cdot 31^{2} \)	\( 3^{4} \cdot 31^{6} \)	$2.58524$	$(-2a+1), (-6a+1)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$31$	31Cs.7.1	$1$	\( 2^{2} \)	$1$	$0.840825582$	3.883607009	\( -12288000 \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -349 a + 110\) , \( -2361 a + 2184\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-349a+110\right){x}-2361a+2184$
77841.3-CMc2	77841.3-CMc	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	77841.3	\( 3^{4} \cdot 31^{2} \)	\( 3^{4} \cdot 31^{6} \)	$2.58524$	$(-2a+1), (6a-5)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$31$	31Cs.7.1	$1$	\( 2^{2} \)	$1$	$0.840825582$	3.883607009	\( -12288000 \)	\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 351 a - 240\) , \( 2011 a + 63\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(351a-240\right){x}+2011a+63$
110889.1-CMc2	110889.1-CMc	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	110889.1	\( 3^{4} \cdot 37^{2} \)	\( 3^{10} \cdot 37^{6} \)	$2.82437$	$(-2a+1), (-7a+4)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$9$	\( 1 \)	$1$	$0.444350091$	4.617821610	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 990 a - 1200\) , \( -15939 a + 11448\bigr] \)	${y}^2+{y}={x}^{3}+\left(990a-1200\right){x}-15939a+11448$
110889.3-CMc2	110889.3-CMc	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	110889.3	\( 3^{4} \cdot 37^{2} \)	\( 3^{10} \cdot 37^{6} \)	$2.82437$	$(-2a+1), (-7a+3)$	0	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓			$9$	\( 1 \)	$1$	$0.444350091$	4.617821610	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -990 a - 210\) , \( 15939 a - 4491\bigr] \)	${y}^2+{y}={x}^{3}+\left(-990a-210\right){x}+15939a-4491$
149769.1-CMb2	149769.1-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	149769.1	\( 3^{4} \cdot 43^{2} \)	\( 3^{4} \cdot 43^{6} \)	$3.04477$	$(-2a+1), (-7a+1)$	$2$	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$43$	43Cs.9.1	$1$	\( 2^{2} \)	$0.354237281$	$0.713924910$	4.672358447	\( -12288000 \)	\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -479 a + 130\) , \( -3378 a + 2952\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-479a+130\right){x}-3378a+2952$
149769.3-CMb2	149769.3-CMb	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	149769.3	\( 3^{4} \cdot 43^{2} \)	\( 3^{4} \cdot 43^{6} \)	$3.04477$	$(-2a+1), (7a-6)$	$2$	$\mathsf{trivial}$	$\textsf{yes}$	$-27$	$\mathrm{U}(1)$	✓			✓	$43$	43Cs.9.1	$1$	\( 2^{2} \)	$0.354237281$	$0.713924910$	4.672358447	\( -12288000 \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 481 a - 350\) , \( 3858 a - 776\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(481a-350\right){x}+3858a-776$

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  *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.