Properties

Label 1309.a.9163.1
Conductor 1309
Discriminant -9163
Mordell-Weil group \(\Z/{8}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = 7x^5 - x^4 - 5x^3 - x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 7x^5z - x^4z^2 - 5x^3z^3 - x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 28x^5 - 3x^4 - 20x^3 - 2x^2 + 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, -5, -1, 7]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, -5, -1, 7], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 4, -2, -20, -3, 28]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1309\) \(=\) \( 7 \cdot 11 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-9163\) \(=\) \( - 7^{2} \cdot 11 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(6784\) \(=\)  \( 2^{7} \cdot 53 \)
\( I_4 \)  \(=\) \(-126464\) \(=\)  \( - 2^{9} \cdot 13 \cdot 19 \)
\( I_6 \)  \(=\) \(-273915456\) \(=\)  \( - 2^{6} \cdot 3 \cdot 1426643 \)
\( I_{10} \)  \(=\) \(-37531648\) \(=\)  \( - 2^{12} \cdot 7^{2} \cdot 11 \cdot 17 \)
\( J_2 \)  \(=\) \(848\) \(=\)  \( 2^{4} \cdot 53 \)
\( J_4 \)  \(=\) \(31280\) \(=\)  \( 2^{4} \cdot 5 \cdot 17 \cdot 23 \)
\( J_6 \)  \(=\) \(1576817\) \(=\)  \( 11 \cdot 29 \cdot 4943 \)
\( J_8 \)  \(=\) \(89675604\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 443 \cdot 5623 \)
\( J_{10} \)  \(=\) \(-9163\) \(=\)  \( - 7^{2} \cdot 11 \cdot 17 \)
\( g_1 \)  \(=\) \(-438509757267968/9163\)
\( g_2 \)  \(=\) \(-1122032353280/539\)
\( g_3 \)  \(=\) \(-103081401088/833\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(8\)

2-torsion field: 6.2.9511568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 13.54561 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.423300 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 7 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 11 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 17 T^{2} )\)

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 \(\Q\).