Elliptic curve search results

Results (13 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
32.4-a1	32.4-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	32.4	\( 2^{5} \)	\( 2^{20} \)	$1.18333$	$(2,a), (2,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 2 \)	$0.139158379$	$8.191609121$	1.637901283	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( a + 2\) , \( -3\bigr] \)	${y}^2={x}^3-{x}^2+\left(a+2\right){x}-3$
256.5-b1	256.5-b	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	256.5	\( 2^{8} \)	\( 2^{20} \)	$1.99012$	$(2,a), (2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$1$	$8.191609121$	2.942512860	\( -1536 a + 2816 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( a + 2\) , \( 3\bigr] \)	${y}^2={x}^3+{x}^2+\left(a+2\right){x}+3$
256.7-a1	256.7-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	256.7	\( 2^{8} \)	\( 2^{26} \)	$1.99012$	$(2,a), (2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$1$	$5.792342358$	2.080670797	\( -1536 a + 2816 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -a - 8\) , \( -a + 8\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-a-8\right){x}-a+8$
512.4-c1	512.4-c	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	512.4	\( 2^{9} \)	\( 2^{14} \)	$2.36667$	$(2,a), (2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 2 \)	$1$	$5.792342358$	4.161341595	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -4\) , \( a\bigr] \)	${y}^2={x}^3-a{x}^2-4{x}+a$
800.10-a1	800.10-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	800.10	\( 2^{5} \cdot 5^{2} \)	\( 2^{8} \cdot 5^{6} \)	$2.64601$	$(2,a), (2,a+1), (5,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 2 \)	$1$	$3.663398968$	2.631863512	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -a + 2\) , \( -2 a - 3\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(-a+2\right){x}-2a-3$
800.12-a1	800.12-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	800.12	\( 2^{5} \cdot 5^{2} \)	\( 2^{20} \cdot 5^{6} \)	$2.64601$	$(2,a), (2,a+1), (5,a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 2 \cdot 3 \)	$0.200541970$	$3.663398968$	3.166794570	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 6 a - 5\) , \( -4 a - 5\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(6a-5\right){x}-4a-5$
1024.5-f1	1024.5-f	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	1024.5	\( 2^{10} \)	\( 2^{14} \)	$2.81446$	$(2,a), (2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$1$	$5.792342358$	2.080670797	\( -1536 a + 2816 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -4\) , \( -a\bigr] \)	${y}^2={x}^3+a{x}^2-4{x}-a$
1024.7-a1	1024.7-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	1024.7	\( 2^{10} \)	\( 2^{26} \)	$2.81446$	$(2,a), (2,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$0.370974314$	$5.792342358$	1.543750847	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -a - 8\) , \( a - 8\bigr] \)	${y}^2={x}^3-a{x}^2+\left(-a-8\right){x}+a-8$
2592.4-a1	2592.4-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	2592.4	\( 2^{5} \cdot 3^{4} \)	\( 2^{20} \cdot 3^{12} \)	$3.55000$	$(2,a), (2,a+1), (3)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 2^{3} \cdot 3 \)	$0.059386400$	$2.730536373$	5.591847865	\( -1536 a + 2816 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 9 a + 15\) , \( -9 a + 65\bigr] \)	${y}^2={x}^3+\left(9a+15\right){x}-9a+65$
3136.13-b1	3136.13-b	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	3136.13	\( 2^{6} \cdot 7^{2} \)	\( 2^{20} \cdot 7^{6} \)	$3.72317$	$(2,a), (2,a+1), (7,a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 3 \)	$0.852028587$	$3.096137224$	5.685579897	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -7 a + 19\) , \( 2 a - 51\bigr] \)	${y}^2={x}^3-{x}^2+\left(-7a+19\right){x}+2a-51$
3136.15-a1	3136.15-a	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	3136.15	\( 2^{6} \cdot 7^{2} \)	\( 2^{8} \cdot 7^{6} \)	$3.72317$	$(2,a), (2,a+1), (7,a+4)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 3 \)	$0.555237745$	$3.096137224$	3.705097000	\( -1536 a + 2816 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3 a - 6\) , \( 2 a - 7\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(3a-6\right){x}+2a-7$
4096.7-c1	4096.7-c	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	4096.7	\( 2^{12} \)	\( 2^{32} \)	$3.98024$	$(2,a), (2,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$0.659464960$	$4.095804560$	1.940484126	\( -1536 a + 2816 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 4 a + 7\) , \( 4 a - 17\bigr] \)	${y}^2={x}^3+{x}^2+\left(4a+7\right){x}+4a-17$
4096.7-d1	4096.7-d	$1$	$1$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	4096.7	\( 2^{12} \)	\( 2^{32} \)	$3.98024$	$(2,a), (2,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2, 5$	2Cn, 5S4	$1$	\( 1 \)	$2.155455004$	$4.095804560$	6.342454071	\( -1536 a + 2816 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a + 7\) , \( -4 a + 17\bigr] \)	${y}^2={x}^3-{x}^2+\left(4a+7\right){x}-4a+17$

 

  *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.