Elliptic curves over \(\Q(\sqrt{65}) \) of conductor 25.1

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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
25.1-a1	25.1-a	$2$	$5$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 2^{12} \cdot 5^{2} \)	$1.61094$	$(5,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$5$	5B.1.4	$1$	\( 1 \)	$1$	$59.51621587$	7.382078040	\( 102400 \)	\( \bigl[0\) , \( a\) , \( a\) , \( 12 a - 48\) , \( -57 a + 256\bigr] \)	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(12a-48\right){x}-57a+256$
25.1-a2	25.1-a	$2$	$5$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 2^{12} \cdot 5^{10} \)	$1.61094$	$(5,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$5$	5B.1.3	$25$	\( 1 \)	$1$	$2.380648635$	7.382078040	\( 27258880 \)	\( \bigl[0\) , \( a\) , \( a\) , \( 642 a - 2928\) , \( 17127 a - 77696\bigr] \)	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(642a-2928\right){x}+17127a-77696$
25.1-b1	25.1-b	$1$	$1$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 5^{10} \)	$1.61094$	$(5,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓			$1$	\( 1 \)	$1$	$7.359657410$	0.912853153	\( 1715 \)	\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -116 a + 534\) , \( -31 a + 140\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-116a+534\right){x}-31a+140$
25.1-c1	25.1-c	$1$	$1$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 2^{12} \cdot 5^{10} \)	$1.61094$	$(5,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓			$1$	\( 1 \)	$1$	$7.359657410$	0.912853153	\( 1715 \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -25 a + 120\) , \( -30 a + 140\bigr] \)	${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-25a+120\right){x}-30a+140$
25.1-d1	25.1-d	$2$	$5$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 5^{2} \)	$1.61094$	$(5,a+2)$	0	$\Z/5\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$5$	5B.1.1	$1$	\( 1 \)	$1$	$59.51621587$	0.295283121	\( 102400 \)	\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 53 a - 236\) , \( -502 a + 2276\bigr] \)	${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(53a-236\right){x}-502a+2276$
25.1-d2	25.1-d	$2$	$5$	\(\Q(\sqrt{65}) \)	$2$	$[2, 0]$	25.1	\( 5^{2} \)	\( 5^{10} \)	$1.61094$	$(5,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$5$	5B.1.2	$1$	\( 1 \)	$1$	$2.380648635$	0.295283121	\( 27258880 \)	\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 2933 a - 13286\) , \( 169196 a - 766648\bigr] \)	${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2933a-13286\right){x}+169196a-766648$

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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.