Elliptic curves over \(\Q(\sqrt{-13}) \) of conductor 576.1

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 (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
576.1-a1	576.1-a	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{8} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{2} \)	$4.760442629$	$4.690728597$	3.096606131	\( \frac{97336}{81} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 8\) , \( 8\bigr] \)	${y}^2={x}^3+{x}^2+8{x}+8$
576.1-a2	576.1-a	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{12} \cdot 3^{4} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$2.380221314$	$4.690728597$	3.096606131	\( \frac{21952}{9} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -2\) , \( 0\bigr] \)	${y}^2={x}^3+{x}^2-2{x}$
576.1-a3	576.1-a	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{24} \cdot 3^{2} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{2} \)	$1.190110657$	$4.690728597$	3.096606131	\( \frac{140608}{3} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -17\) , \( -33\bigr] \)	${y}^2={x}^3+{x}^2-17{x}-33$
576.1-a4	576.1-a	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{2} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 1 \)	$4.760442629$	$4.690728597$	3.096606131	\( \frac{7301384}{3} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -32\) , \( 60\bigr] \)	${y}^2={x}^3+{x}^2-32{x}+60$
576.1-b1	576.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{8} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{2} \)	$2.166478915$	$4.690728597$	5.637065639	\( \frac{97336}{81} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 8\) , \( -8\bigr] \)	${y}^2={x}^3-{x}^2+8{x}-8$
576.1-b2	576.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{12} \cdot 3^{4} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$4.332957830$	$4.690728597$	5.637065639	\( \frac{21952}{9} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -2\) , \( 0\bigr] \)	${y}^2={x}^3-{x}^2-2{x}$
576.1-b3	576.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{24} \cdot 3^{2} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$4$	\( 2^{2} \)	$2.166478915$	$4.690728597$	5.637065639	\( \frac{140608}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -17\) , \( 33\bigr] \)	${y}^2={x}^3-{x}^2-17{x}+33$
576.1-b4	576.1-b	$4$	$4$	\(\Q(\sqrt{-13}) \)	$2$	$[0, 1]$	576.1	\( 2^{6} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{2} \)	$3.15679$	$(2,a+1), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 1 \)	$8.665915661$	$4.690728597$	5.637065639	\( \frac{7301384}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -32\) , \( -60\bigr] \)	${y}^2={x}^3-{x}^2-32{x}-60$

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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.