Elliptic curves over \(\Q(\sqrt{269}) \)

Results (5 matches)

 

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
9.1-a1	9.1-a	$1$	$1$	\(\Q(\sqrt{269}) \)	$2$	$[2, 0]$	9.1	\( 3^{2} \)	\( 3^{4} \)	$2.53849$	$(3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2$	2Cn	$1$	\( 2 \)	$1$	$46.51073359$	5.671618953	\( \frac{32768}{9} \)	\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 1093 a - 9490\) , \( -44015 a + 382974\bigr] \)	${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1093a-9490\right){x}-44015a+382974$
11.1-a1	11.1-a	$1$	$1$	\(\Q(\sqrt{269}) \)	$2$	$[2, 0]$	11.1	\( 11 \)	\( 11^{5} \)	$2.66909$	$(-a+8)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$4$	\( 1 \)	$1$	$4.226806287$	1.030851710	\( \frac{109934875}{161051} a - \frac{956522250}{161051} \)	\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 12 a - 36\) , \( 189 a - 1524\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(12a-36\right){x}+189a-1524$
11.1-b1	11.1-b	$1$	$1$	\(\Q(\sqrt{269}) \)	$2$	$[2, 0]$	11.1	\( 11 \)	\( 11 \)	$2.66909$	$(-a+8)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$1.134449587$	$15.55158977$	4.302727522	\( \frac{1459}{11} a + \frac{5087}{11} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -3 a + 85\) , \( -6 a + 150\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a+85\right){x}-6a+150$
11.2-a1	11.2-a	$1$	$1$	\(\Q(\sqrt{269}) \)	$2$	$[2, 0]$	11.2	\( 11 \)	\( 11^{5} \)	$2.66909$	$(-a-7)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$4$	\( 1 \)	$1$	$4.226806287$	1.030851710	\( -\frac{109934875}{161051} a - \frac{846587375}{161051} \)	\( \bigl[a\) , \( 0\) , \( 1\) , \( -13 a - 23\) , \( -189 a - 1335\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(-13a-23\right){x}-189a-1335$
11.2-b1	11.2-b	$1$	$1$	\(\Q(\sqrt{269}) \)	$2$	$[2, 0]$	11.2	\( 11 \)	\( 11 \)	$2.66909$	$(-a-7)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$1.134449587$	$15.55158977$	4.302727522	\( -\frac{1459}{11} a + \frac{6546}{11} \)	\( \bigl[a\) , \( -1\) , \( 1\) , \( 2 a + 83\) , \( 6 a + 144\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(2a+83\right){x}+6a+144$

 

  *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.