Elliptic curves over \(\Q(\sqrt{-10}) \) of conductor 40.1

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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
40.1-a1	40.1-a	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{8} \cdot 5^{8} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{3} \)	$1$	$2.996888981$	1.895399015	\( \frac{237276}{625} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( 9\) , \( 5\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+9{x}+5$
40.1-a2	40.1-a	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{4} \cdot 5^{4} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{2} \)	$1$	$5.993777963$	1.895399015	\( \frac{148176}{25} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( 4\) , \( 4\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+4{x}+4$
40.1-a3	40.1-a	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{20} \cdot 5^{2} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{2} \)	$1$	$5.993777963$	1.895399015	\( \frac{55296}{5} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -8\) , \( -8\bigr] \)	${y}^2={x}^3-8{x}-8$
40.1-a4	40.1-a	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{8} \cdot 5^{2} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{3} \)	$1$	$2.996888981$	1.895399015	\( \frac{132304644}{5} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -21\) , \( 69\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-21{x}+69$
40.1-b1	40.1-b	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{20} \cdot 5^{8} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{4} \)	$1$	$2.996888981$	0.947699507	\( \frac{237276}{625} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 13\) , \( -34\bigr] \)	${y}^2={x}^3+13{x}-34$
40.1-b2	40.1-b	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{16} \cdot 5^{4} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{3} \)	$1$	$5.993777963$	0.947699507	\( \frac{148176}{25} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( -6\bigr] \)	${y}^2={x}^3-7{x}-6$
40.1-b3	40.1-b	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{8} \cdot 5^{2} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{3} \)	$1$	$5.993777963$	0.947699507	\( \frac{55296}{5} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 1\bigr] \)	${y}^2={x}^3-2{x}+1$
40.1-b4	40.1-b	$4$	$4$	\(\Q(\sqrt{-10}) \)	$2$	$[0, 1]$	40.1	\( 2^{3} \cdot 5 \)	\( 2^{20} \cdot 5^{2} \)	$1.42129$	$(2,a), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{2} \)	$1$	$2.996888981$	0.947699507	\( \frac{132304644}{5} \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -107\) , \( -426\bigr] \)	${y}^2={x}^3-107{x}-426$

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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.