Properties

Label 258357.a.258357.1
Conductor $258357$
Discriminant $258357$
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^2y = x^5 - 22x^4 - 30x^3 + x^2 + 11x + 2$ (homogenize, simplify)
$y^2 + x^2zy = x^5z - 22x^4z^2 - 30x^3z^3 + x^2z^4 + 11xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 87x^4 - 120x^3 + 4x^2 + 44x + 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 11, 1, -30, -22, 1]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 11, 1, -30, -22, 1], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([8, 44, 4, -120, -87, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(258357\) \(=\) \( 3 \cdot 11 \cdot 7829 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(258357\) \(=\) \( 3 \cdot 11 \cdot 7829 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(24752\) \(=\)  \( 2^{4} \cdot 7 \cdot 13 \cdot 17 \)
\( I_4 \)  \(=\) \(15988612\) \(=\)  \( 2^{2} \cdot 67 \cdot 59659 \)
\( I_6 \)  \(=\) \(110693925423\) \(=\)  \( 3^{2} \cdot 229 \cdot 53708843 \)
\( I_{10} \)  \(=\) \(1033428\) \(=\)  \( 2^{2} \cdot 3 \cdot 11 \cdot 7829 \)
\( J_2 \)  \(=\) \(12376\) \(=\)  \( 2^{3} \cdot 7 \cdot 13 \cdot 17 \)
\( J_4 \)  \(=\) \(3717122\) \(=\)  \( 2 \cdot 19 \cdot 23 \cdot 4253 \)
\( J_6 \)  \(=\) \(1249461841\) \(=\)  \( 61 \cdot 20482981 \)
\( J_8 \)  \(=\) \(411585945333\) \(=\)  \( 3 \cdot 167 \cdot 821528833 \)
\( J_{10} \)  \(=\) \(258357\) \(=\)  \( 3 \cdot 11 \cdot 7829 \)
\( g_1 \)  \(=\) \(290336410647019749376/258357\)
\( g_2 \)  \(=\) \(7046082395391183872/258357\)
\( g_3 \)  \(=\) \(191374292674417216/258357\)

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

magma: [C![1,0,0]]; // minimal model
 
magma: [C![1,0,0]]; // simplified model
 

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + xz - 2z^2\) \(=\) \(0,\) \(8y\) \(=\) \(xz^2 - 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + xz - 2z^2\) \(=\) \(0,\) \(8y\) \(=\) \(xz^2 - 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + xz - 2z^2\) \(=\) \(0,\) \(8y\) \(=\) \(x^2z + 2xz^2 - 4z^3\) \(0\) \(2\)

2-torsion field: 6.6.35243123229072.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 1.766348 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 3.974284 \)
Analytic order of Ш: \( 9 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 3 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 11 T^{2} )\)
\(7829\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 94 T + 7829 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);