Elliptic curves over \(\Q(\sqrt{-31}) \) of conductor 729.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over imaginary quadratic fields with absolute discriminant 31

Note: The completeness Only modular elliptic curves are included

Results (8 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
729.1-a1	729.1-a	$2$	$3$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 2^{12} \cdot 3^{18} \)	$2.58524$	$(3)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$3$	3B.1.2	$1$	\( 1 \)	$1.403979249$	$2.702876088$	2.726251827	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -7 a + 56\bigr] \)	${y}^2+a{y}={x}^3-7a+56$
729.1-a2	729.1-a	$2$	$3$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 2^{12} \cdot 3^{6} \)	$2.58524$	$(3)$	$1$	$\Z/3\Z$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$3$	3B.1.1	$1$	\( 1 \)	$4.211937749$	$8.108628264$	2.726251827	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 0\bigr] \)	${y}^2+a{y}={x}^3$
729.1-b1	729.1-b	$4$	$27$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 3^{10} \)	$2.58524$	$(3)$	$2$	$\Z/3\Z$	$\textsf{potential}$	$-27$	$N(\mathrm{U}(1))$	✓	✓		✓	$3$	3B.1.1	$1$	\( 3 \)	$3.009274303$	$2.702876088$	3.895612925	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \)	${y}^2+{y}={x}^3-30{x}+63$
729.1-b2	729.1-b	$4$	$27$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 3^{22} \)	$2.58524$	$(3)$	$2$	$\mathsf{trivial}$	$\textsf{potential}$	$-27$	$N(\mathrm{U}(1))$	✓	✓		✓	$3$	3B.1.2	$1$	\( 1 \)	$3.009274303$	$0.900958696$	3.895612925	\( -12288000 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( -270\) , \( -1708\bigr] \)	${y}^2+{y}={x}^3-270{x}-1708$
729.1-b3	729.1-b	$4$	$27$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 3^{18} \)	$2.58524$	$(3)$	$2$	$\Z/3\Z$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓	✓		✓	$3$	3Cs.1.1	$1$	\( 3 \)	$3.009274303$	$2.702876088$	3.895612925	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -7\bigr] \)	${y}^2+{y}={x}^3-7$
729.1-b4	729.1-b	$4$	$27$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 3^{6} \)	$2.58524$	$(3)$	$2$	$\Z/3\Z$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓	✓		✓	$3$	3Cs.1.1	$1$	\( 1 \)	$3.009274303$	$8.108628264$	3.895612925	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \)	${y}^2+{y}={x}^3$
729.1-c1	729.1-c	$2$	$3$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 2^{12} \cdot 3^{18} \)	$2.58524$	$(3)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$3$	3B.1.2	$1$	\( 1 \)	$1.403979249$	$2.702876088$	2.726251827	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( 6 a + 49\bigr] \)	${y}^2+\left(a+1\right){y}={x}^3+6a+49$
729.1-c2	729.1-c	$2$	$3$	\(\Q(\sqrt{-31}) \)	$2$	$[0, 1]$	729.1	\( 3^{6} \)	\( 2^{12} \cdot 3^{6} \)	$2.58524$	$(3)$	$1$	$\Z/3\Z$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$3$	3B.1.1	$1$	\( 1 \)	$4.211937749$	$8.108628264$	2.726251827	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -a\bigr] \)	${y}^2+\left(a+1\right){y}={x}^3-a$

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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.