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Properties

Label 100325.b.100325.1
Conductor $100325$
Discriminant $100325$
Mordell-Weil group \(\Z\)
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

$y^2 + (x^2 + x + 1)y = -x^5 + 3x^4 - 10x^2 + 2x + 8$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = -x^5z + 3x^4z^2 - 10x^2z^4 + 2xz^5 + 8z^6$ (dehomogenize, simplify)
$y^2 = -4x^5 + 13x^4 + 2x^3 - 37x^2 + 10x + 33$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 2, -10, 0, 3, -1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![8, 2, -10, 0, 3, -1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([33, 10, -37, 2, 13, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(100325\) \(=\) \( 5^{2} \cdot 4013 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(100325\) \(=\) \( 5^{2} \cdot 4013 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3060\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
\( I_4 \)  \(=\) \(-200871\) \(=\)  \( - 3^{2} \cdot 11 \cdot 2029 \)
\( I_6 \)  \(=\) \(-238360131\) \(=\)  \( - 3^{3} \cdot 503 \cdot 17551 \)
\( I_{10} \)  \(=\) \(12841600\) \(=\)  \( 2^{7} \cdot 5^{2} \cdot 4013 \)
\( J_2 \)  \(=\) \(765\) \(=\)  \( 3^{2} \cdot 5 \cdot 17 \)
\( J_4 \)  \(=\) \(32754\) \(=\)  \( 2 \cdot 3 \cdot 53 \cdot 103 \)
\( J_6 \)  \(=\) \(2568348\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 7927 \)
\( J_8 \)  \(=\) \(222990426\) \(=\)  \( 2 \cdot 3^{2} \cdot 101 \cdot 173 \cdot 709 \)
\( J_{10} \)  \(=\) \(100325\) \(=\)  \( 5^{2} \cdot 4013 \)
\( g_1 \)  \(=\) \(10480141999125/4013\)
\( g_2 \)  \(=\) \(586554865290/4013\)
\( g_3 \)  \(=\) \(60122458332/4013\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (2 : -3 : 1),\, (2 : -4 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (-1 : -1 : 1),\, (2 : -3 : 1),\, (2 : -4 : 1)\)
All points: \((1 : 0 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (2 : -1 : 1),\, (2 : 1 : 1)\)

magma: [C![-1,-1,1],C![-1,0,1],C![1,0,0],C![2,-4,1],C![2,-3,1]]; // minimal model
 
magma: [C![-1,-1,1],C![-1,1,1],C![1,0,0],C![2,-1,1],C![2,1,1]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - 2z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.093538\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - 2z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.093538\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) + (2 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - 2z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 - z^3\) \(0.093538\) \(\infty\)

2-torsion field: 5.1.1605200.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.093538 \)
Real period: \( 13.29280 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.243384 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(4013\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 83 T + 4013 T^{2} )\)

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?
\(2\) 2.6.1 no

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);