Elliptic curves over \(\Q(\sqrt{3}) \) of conductor 64.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over real quadratic fields with discriminant 497

Results (4 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	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	64.1	\( 2^{6} \)	\( 2^{12} \)	$0.87554$	$(a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 2 \)	$1$	$27.50074327$	0.992347595	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \)	${y}^2={x}^{3}-{x}$
64.1-a2	64.1-a	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	64.1	\( 2^{6} \)	\( 2^{12} \)	$0.87554$	$(a+1)$	0	$\Z/4\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$1$	$13.75037163$	0.992347595	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 a + 7\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-4a+7\right){x}$
64.1-a3	64.1-a	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	64.1	\( 2^{6} \)	\( 2^{6} \)	$0.87554$	$(a+1)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 1 \)	$1$	$13.75037163$	0.992347595	\( 287496 \)	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 11 a - 17\) , \( 17 a - 29\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(11a-17\right){x}+17a-29$
64.1-a4	64.1-a	$4$	$4$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	64.1	\( 2^{6} \)	\( 2^{6} \)	$0.87554$	$(a+1)$	0	$\Z/4\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 1 \)	$1$	$55.00148654$	0.992347595	\( 287496 \)	\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 10 a - 19\) , \( -36 a + 61\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-19\right){x}-36a+61$

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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.