Genus 2 curves of conductor 997

Results (3 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
997.a.997.1	997.a	\( 997 \)	\( 997 \)	$0$	$2$	$\Z/2\Z\oplus\Z/4\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2$	✓	✓	$C_2$	$C_2$	$3$	$3$	2.120.3	✓	✓	$1$	\( 1 \)	\(1.000000\)	\(21.589621\)	\(0.337338\)	$[6112,48064,98113399,3988]$	$[3056,381120,61964417,11027700988,997]$	$[266542673508171776/997,10877317101649920/997,578694117523712/997]$	$y^2 + xy = x^5 - 8x^4 + 16x^3 - x$
997.a.997.2	997.a	\( 997 \)	\( 997 \)	$0$	$1$	$\Z/8\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2$	✓	✓	$C_2$	$C_2$	$4$	$2$	2.30.3	✓	✓	$1$	\( 1 \)	\(1.000000\)	\(21.589621\)	\(0.337338\)	$[64,184,391,3988]$	$[32,12,305,2404,997]$	$[33554432/997,393216/997,312320/997]$	$y^2 + (x + 1)y = x^5 + x^4$
997.b.997.1	997.b	\( 997 \)	\( 997 \)	$1$	$1$	$\Z/3\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2,3$	✓	✓	$C_2$	$C_2$	$7$	$1$	2.6.1, 3.80.1	✓	✓	$1$	\( 1 \)	\(0.081270\)	\(19.932843\)	\(0.179992\)	$[32,16,-1680,-3988]$	$[16,8,208,816,-997]$	$[-1048576/997,-32768/997,-53248/997]$	$y^2 + y = x^5 - 2x^4 + 2x^3 - x^2$