Genus 2 curves of conductor 688

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
688.a.2752.1	688.a	\( 2^{4} \cdot 43 \)	\( - 2^{6} \cdot 43 \)	$0$	$1$	$\Z/20\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2,5$	✓	✓	$C_2$	$C_2$	$6$	$2$	2.30.3	✓	✓	$1$	\( 5 \)	\(1.000000\)	\(25.707298\)	\(0.321341\)	$[32,112,-680,-344]$	$[32,-32,1344,10496,-2752]$	$[-524288/43,16384/43,-21504/43]$	$y^2 + y = 2x^5 - 5x^4 + 4x^3 - x$
688.a.704512.2	688.a	\( 2^{4} \cdot 43 \)	\( - 2^{14} \cdot 43 \)	$0$	$1$	$\Z/10\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2,5$	✓	✓	$C_2$	$C_2$	$4$	$2$	2.30.3	✓	✓	$1$	\( 5 \)	\(1.000000\)	\(6.426825\)	\(0.321341\)	$[464,-248,-39602,-86]$	$[1856,146176,15688704,1937702912,-704512]$	$[-1344218660864/43,-57041383424/43,-3298550016/43]$	$y^2 = 2x^5 - 7x^4 - 8x^3 + 2x^2 + 4x + 1$
688.a.704512.1	688.a	\( 2^{4} \cdot 43 \)	\( 2^{14} \cdot 43 \)	$0$	$1$	$\Z/10\Z$	\(\Q\)	\(\Q\)		$\mathrm{USp}(4)$	$2,5$	✓	✓	$C_2$	$C_2$	$3$	$1$	2.60.1	✓	✓	$1$	\( 5 \)	\(1.000000\)	\(6.426825\)	\(0.321341\)	$[128,532,26830,86]$	$[512,5248,-408576,-59183104,704512]$	$[2147483648/43,42991616/43,-6537216/43]$	$y^2 = 2x^5 + 4x^3 + x^2 + 2x + 1$