Properties

Label 100261.a.100261.1
Conductor 100261
Discriminant -100261
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 7, -1, -16, 1], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 7, -1, -16, 1]), R([1, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 7, -1, -16, 1], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -10, 29, -4, -64, 4]))
 

$y^2 + (x + 1)y = x^5 - 16x^4 - x^3 + 7x^2 - 3x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 16x^4z^2 - x^3z^3 + 7x^2z^4 - 3xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 64x^4 - 4x^3 + 29x^2 - 10x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(100261\) = \( 7 \cdot 14323 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-100261\) = \( - 7 \cdot 14323 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(28192\) =  \( 2^{5} \cdot 881 \)
\( I_4 \)  = \(-507776\) =  \( - 2^{7} \cdot 3967 \)
\( I_6 \)  = \(-4691938624\) =  \( - 2^{6} \cdot 37 \cdot 1981393 \)
\( I_{10} \)  = \(-410669056\) =  \( - 2^{12} \cdot 7 \cdot 14323 \)
\( J_2 \)  = \(3524\) =  \( 2^{2} \cdot 881 \)
\( J_4 \)  = \(522730\) =  \( 2 \cdot 5 \cdot 13 \cdot 4021 \)
\( J_6 \)  = \(104271441\) =  \( 3 \cdot 907 \cdot 38321 \)
\( J_8 \)  = \(23551476296\) =  \( 2^{3} \cdot 19 \cdot 2017 \cdot 76819 \)
\( J_{10} \)  = \(-100261\) =  \( - 7 \cdot 14323 \)
\( g_1 \)  = \(-543474909254042624/100261\)
\( g_2 \)  = \(-22876265307259520/100261\)
\( g_3 \)  = \(-1294902814688016/100261\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![0,-1,1],C![0,0,1],C![1,-42,4],C![1,-38,4],C![1,0,0],C![21,-994,1],C![21,972,1]];
 

Known points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (1 : -38 : 4),\, (1 : -42 : 4),\, (21 : 972 : 1),\, (21 : -994 : 1)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(1\)

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z \times \Z\)

Generator Height Order
\((-4x + z) x\) \(=\) \(0,\) \(8y\) \(=\) \(11xz^2 - 8z^3\) \(1.274501\) \(\infty\)
\(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.170227\) \(\infty\)

2-torsion field: 5.3.1604176.1

BSD invariants

Analytic rank: \(2\)   (upper bound)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.215449 \)
Real period: \( 6.839751 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.473618 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 7 T^{2} )\)
\(14323\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 140 T + 14323 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).