Elliptic curves over \(\Q(\sqrt{10}) \) of conductor 260.2

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 (6 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
260.2-a1	260.2-a	$1$	$1$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{20} \cdot 5 \cdot 13 \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 1 \)	$1$	$5.496886437$	0.869134059	\( -\frac{42496}{65} a - \frac{26880}{13} \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( 8 a - 22\) , \( 1854 a - 5864\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(8a-22\right){x}+1854a-5864$
260.2-b1	260.2-b	$2$	$3$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{20} \cdot 5^{3} \cdot 13^{3} \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	\( 3^{2} \)	$1$	$1.504701855$	2.141228278	\( -\frac{229616225792}{54925} a - \frac{145218683136}{10985} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -72 a + 215\) , \( 1240 a - 3935\bigr] \)	${y}^2={x}^{3}-{x}^{2}+\left(-72a+215\right){x}+1240a-3935$
260.2-b2	260.2-b	$2$	$3$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{20} \cdot 5 \cdot 13 \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	\( 1 \)	$1$	$13.54231669$	2.141228278	\( \frac{3584}{65} a - \frac{2304}{13} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 8 a - 25\) , \( -56 a + 177\bigr] \)	${y}^2={x}^{3}-{x}^{2}+\left(8a-25\right){x}-56a+177$
260.2-c1	260.2-c	$2$	$3$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{8} \cdot 5^{3} \cdot 13^{3} \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	\( 3 \)	$1$	$1.504701855$	0.713742759	\( -\frac{229616225792}{54925} a - \frac{145218683136}{10985} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -18 a + 54\) , \( 146 a - 465\bigr] \)	${y}^2={x}^{3}+{x}^{2}+\left(-18a+54\right){x}+146a-465$
260.2-c2	260.2-c	$2$	$3$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{8} \cdot 5 \cdot 13 \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	\( 3 \)	$1$	$13.54231669$	0.713742759	\( \frac{3584}{65} a - \frac{2304}{13} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 2 a - 6\) , \( -6 a + 19\bigr] \)	${y}^2={x}^{3}+{x}^{2}+\left(2a-6\right){x}-6a+19$
260.2-d1	260.2-d	$1$	$1$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	260.2	\( 2^{2} \cdot 5 \cdot 13 \)	\( 2^{8} \cdot 5 \cdot 13 \)	$2.26940$	$(2,a), (5,a), (13,a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 3 \)	$1$	$5.496886437$	2.607402177	\( -\frac{42496}{65} a - \frac{26880}{13} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 2 a - 3\) , \( 229 a - 723\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(2a-3\right){x}+229a-723$

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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.