Genus 2 curves of conductor 405769

Results (displaying both matches)

 

Label	Class	Conductor	Discriminant	Rank*	2-Selmer rank	Torsion	$\textrm{End}^0(J_{\overline\Q})$	$\textrm{End}^0(J)$	$\GL_2\textsf{-type}$	Sato-Tate	Nonmaximal primes	$\Q$-simple	\(\overline{\Q}\)-simple	\(\Aut(X)\)	\(\Aut(X_{\overline{\Q}})\)	$\Q$-points	$\Q$-Weierstrass points	mod-$\ell$ images	Locally solvable	Square Ш*	Analytic Ш*	Tamagawa	Regulator	Real period	Leading coefficient	Igusa-Clebsch invariants	Igusa invariants	G2-invariants	Equation
405769.a.405769.1	405769.a	\( 7^{4} \cdot 13^{2} \)	\( 7^{4} \cdot 13^{2} \)	$2$	$2$	$\mathsf{trivial}$	\(\mathsf{RM}\)	\(\mathsf{RM}\)	✓	$\mathrm{SU}(2)\times\mathrm{SU}(2)$		✓	✓	$C_2$	$C_2$	$4$	$0$	2.60.4, 3.2160.18	✓	✓	$1$	\( 1 \)	\(0.191488\)	\(9.244757\)	\(1.770260\)	$[388,1225,723093,51938432]$	$[97,341,-6555,-188029,405769]$	$[8587340257/405769,311221493/405769,-61675995/405769]$	$y^2 + (x^3 + x + 1)y = 2x^5 + 2x^4 + 3x^3 + x^2$
405769.b.405769.1	405769.b	\( 7^{4} \cdot 13^{2} \)	\( 7^{4} \cdot 13^{2} \)	$0$	$0$	$\mathsf{trivial}$	\(\mathrm{M}_2(\Q)\)	\(\mathsf{CM}\)	✓	$E_6$		✓		$C_6$	$D_6$	$0$	$0$	2.80.1, 3.480.12		✓	$1$	\( 1 \)	\(1.000000\)	\(6.383879\)	\(6.383879\)	$[2596,375193,248614093,51938432]$	$[649,1917,-1907,-1228133,405769]$	$[115139273278249/405769,524030063733/405769,-803230307/405769]$	$y^2 + (x^3 + x + 1)y = -x^6 + 4x^5 - 5x^4 - 4x^3 + 4x^2 - 1$