Elliptic curves over \(\Q(\sqrt{-13}) \) of conductor 64.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 1000 over imaginary quadratic fields with absolute discriminant 52

Note: The completeness Only modular elliptic curves are included

Results (6 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
64.1-a1	64.1-a	$1$	$1$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{6} \)	$1.82257$	$(2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Cn	$1$	\( 1 \)	$1$	$7.681576726$	2.130486058	\( -74088 \)	\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -4 a + 1\) , \( 10\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-4a+1\right){x}+10$
64.1-b1	64.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{12} \)	$1.82257$	$(2,a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 2 \)	$5.832511736$	$6.875185818$	2.780407135	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \)	${y}^2={x}^3-{x}$
64.1-b2	64.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{24} \)	$1.82257$	$(2,a+1)$	$1$	$\Z/4\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 2^{2} \)	$2.916255868$	$6.875185818$	2.780407135	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( 0\bigr] \)	${y}^2={x}^3+4{x}$
64.1-b3	64.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{18} \)	$1.82257$	$(2,a+1)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 1 \)	$2.916255868$	$6.875185818$	2.780407135	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( -14\bigr] \)	${y}^2={x}^3-11{x}-14$
64.1-b4	64.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{18} \)	$1.82257$	$(2,a+1)$	$1$	$\Z/4\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 1 \)	$11.66502347$	$6.875185818$	2.780407135	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( 14\bigr] \)	${y}^2={x}^3-11{x}+14$
64.1-c1	64.1-c	$1$	$1$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	64.1	\( 2^{6} \)	\( 2^{6} \)	$1.82257$	$(2,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Cn	$1$	\( 1 \)	$1$	$7.681576726$	2.130486058	\( -74088 \)	\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -3 a - 5\) , \( a + 7\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^3+a{x}^2+\left(-3a-5\right){x}+a+7$

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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.