Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 6561.1

Results (8 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
6561.1-CMf1	6561.1-CMf	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -3 a\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}-3a$
6561.1-CMe1	6561.1-CMe	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 2 a - 2\bigr] \)	${y}^2+a{y}={x}^{3}+2a-2$
6561.1-CMd1	6561.1-CMd	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓	✓		✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 2\bigr] \)	${y}^2+{y}={x}^{3}+2$
6561.1-CMc1	6561.1-CMc	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$5.622208826$	2.163989186	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( 0\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}$
6561.1-CMb1	6561.1-CMb	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$5.622208826$	2.163989186	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a + 1\bigr] \)	${y}^2+a{y}={x}^{3}-a+1$
6561.1-CMa1	6561.1-CMa	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	$2$	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓	✓		✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$0.192305923$	$5.622208826$	1.664591759	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -1\bigr] \)	${y}^2+{y}={x}^{3}-1$
6561.1-a1	6561.1-a	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cn, 3B.1.1[2]	$1$	\( 3 \)	$1$	$3.153971350$	1.213964138	\( -33507 a - 11637 \)	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 4 a - 11\) , \( 6 a - 10\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-11\right){x}+6a-10$
6561.1-a2	6561.1-a	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cn, 3B.1.1[2]	$1$	\( 3 \)	$1$	$3.153971350$	1.213964138	\( 33507 a - 45144 \)	\( \bigl[1\) , \( -1\) , \( a\) , \( -5 a - 6\) , \( -7 a - 3\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-5a-6\right){x}-7a-3$

 

  *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.