Properties

Label 50608.b.809728.1
Conductor $50608$
Discriminant $-809728$
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

$y^2 = x^5 - 30x^3 - 72x^2 - 28x - 3$ (homogenize, simplify)
$y^2 = x^5z - 30x^3z^3 - 72x^2z^4 - 28xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = x^5 - 30x^3 - 72x^2 - 28x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -28, -72, -30, 0, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -28, -72, -30, 0, 1], R![]);
 
sage: X = HyperellipticCurve(R([-3, -28, -72, -30, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(50608\) \(=\) \( 2^{4} \cdot 3163 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-809728\) \(=\) \( - 2^{8} \cdot 3163 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(4280\) \(=\)  \( 2^{3} \cdot 5 \cdot 107 \)
\( I_4 \)  \(=\) \(961120\) \(=\)  \( 2^{5} \cdot 5 \cdot 6007 \)
\( I_6 \)  \(=\) \(1057109684\) \(=\)  \( 2^{2} \cdot 37 \cdot 7142633 \)
\( I_{10} \)  \(=\) \(-3163\) \(=\)  \( -3163 \)
\( J_2 \)  \(=\) \(8560\) \(=\)  \( 2^{4} \cdot 5 \cdot 107 \)
\( J_4 \)  \(=\) \(490080\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \cdot 1021 \)
\( J_6 \)  \(=\) \(28891136\) \(=\)  \( 2^{11} \cdot 14107 \)
\( J_8 \)  \(=\) \(1782429440\) \(=\)  \( 2^{8} \cdot 5 \cdot 11 \cdot 71 \cdot 1783 \)
\( J_{10} \)  \(=\) \(-809728\) \(=\)  \( - 2^{8} \cdot 3163 \)
\( g_1 \)  \(=\) \(-179526621529600000/3163\)
\( g_2 \)  \(=\) \(-1200738146880000/3163\)
\( g_3 \)  \(=\) \(-8269365401600/3163\)

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: \((1 : 0 : 0),\, (-3 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (-3 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (-3 : 0 : 1)\)

magma: [C![-3,0,1],C![1,0,0]]; // minimal model
 
magma: [C![-3,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

2-torsion field: 4.2.3163.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.891558 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.006006 \)
Analytic order of Ш: \( 9 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(8\) \(1\) \(1\)
\(3163\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 76 T + 3163 T^{2} )\)

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?
\(2\) 2.30.3 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);