Elliptic curves over \(\Q(\sqrt{-1}) \) of conductor 512.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100000 over imaginary quadratic fields with absolute discriminant 4

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
512.1-a1	512.1-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{17} \)	$0.85013$	$(a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 1 \)	$1$	$3.920761445$	0.980190361	\( -118792 a - 11528 \)	\( \bigl[0\) , \( -i + 1\) , \( 0\) , \( -6 i + 5\) , \( -i + 7\bigr] \)	${y}^2={x}^{3}+\left(-i+1\right){x}^{2}+\left(-6i+5\right){x}-i+7$
512.1-a2	512.1-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{17} \)	$0.85013$	$(a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$3.920761445$	0.980190361	\( 118792 a - 11528 \)	\( \bigl[0\) , \( i - 1\) , \( 0\) , \( -6 i - 5\) , \( -7 i + 1\bigr] \)	${y}^2={x}^{3}+\left(i-1\right){x}^{2}+\left(-6i-5\right){x}-7i+1$
512.1-a3	512.1-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{10} \)	$0.85013$	$(a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2 \)	$1$	$7.841522890$	0.980190361	\( 128 \)	\( \bigl[0\) , \( i - 1\) , \( 0\) , \( -i\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(i-1\right){x}^{2}-i{x}$
512.1-a4	512.1-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{20} \)	$0.85013$	$(a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$3.920761445$	0.980190361	\( 10976 \)	\( \bigl[0\) , \( i - 1\) , \( 0\) , \( 4 i\) , \( -4 i - 4\bigr] \)	${y}^2={x}^{3}+\left(i-1\right){x}^{2}+4i{x}-4i-4$
512.1-b1	512.1-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{17} \)	$0.85013$	$(a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$3.920761445$	0.980190361	\( -118792 a - 11528 \)	\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( 6 i - 5\) , \( 7 i + 1\bigr] \)	${y}^2={x}^{3}+\left(-i-1\right){x}^{2}+\left(6i-5\right){x}+7i+1$
512.1-b2	512.1-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{17} \)	$0.85013$	$(a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 1 \)	$1$	$3.920761445$	0.980190361	\( 118792 a - 11528 \)	\( \bigl[0\) , \( i + 1\) , \( 0\) , \( 6 i + 5\) , \( i + 7\bigr] \)	${y}^2={x}^{3}+\left(i+1\right){x}^{2}+\left(6i+5\right){x}+i+7$
512.1-b3	512.1-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{10} \)	$0.85013$	$(a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2 \)	$1$	$7.841522890$	0.980190361	\( 128 \)	\( \bigl[0\) , \( i + 1\) , \( 0\) , \( i\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(i+1\right){x}^{2}+i{x}$
512.1-b4	512.1-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	512.1	\( 2^{9} \)	\( 2^{20} \)	$0.85013$	$(a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$3.920761445$	0.980190361	\( 10976 \)	\( \bigl[0\) , \( i + 1\) , \( 0\) , \( -4 i\) , \( -4 i + 4\bigr] \)	${y}^2={x}^{3}+\left(i+1\right){x}^{2}-4i{x}-4i+4$

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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.