Properties

Label 2735.a.13675.1
Conductor $2735$
Discriminant $13675$
Mordell-Weil group trivial
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

$y^2 + (x^2 + x + 1)y = x^5 - x^4 - 39x^3 - 76x^2 + 24x - 6$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = x^5z - x^4z^2 - 39x^3z^3 - 76x^2z^4 + 24xz^5 - 6z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 3x^4 - 154x^3 - 301x^2 + 98x - 23$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 24, -76, -39, -1, 1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 24, -76, -39, -1, 1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([-23, 98, -301, -154, -3, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2735\) \(=\) \( 5 \cdot 547 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(13675\) \(=\) \( 5^{2} \cdot 547 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(71764\) \(=\)  \( 2^{2} \cdot 7 \cdot 11 \cdot 233 \)
\( I_4 \)  \(=\) \(259832017\) \(=\)  \( 157 \cdot 1654981 \)
\( I_6 \)  \(=\) \(4804066188809\) \(=\)  \( 191 \cdot 61703 \cdot 407633 \)
\( I_{10} \)  \(=\) \(1750400\) \(=\)  \( 2^{7} \cdot 5^{2} \cdot 547 \)
\( J_2 \)  \(=\) \(17941\) \(=\)  \( 7 \cdot 11 \cdot 233 \)
\( J_4 \)  \(=\) \(2585311\) \(=\)  \( 19 \cdot 136069 \)
\( J_6 \)  \(=\) \(598781761\) \(=\)  \( 598781761 \)
\( J_8 \)  \(=\) \(1014727651845\) \(=\)  \( 3 \cdot 5 \cdot 67648510123 \)
\( J_{10} \)  \(=\) \(13675\) \(=\)  \( 5^{2} \cdot 547 \)
\( g_1 \)  \(=\) \(1858802427581887565701/13675\)
\( g_2 \)  \(=\) \(14929756777053326131/13675\)
\( g_3 \)  \(=\) \(192735562462946041/13675\)

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 5.1.8752.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.310710 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.621420 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 - 3 T + 5 T^{2} )\)
\(547\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 547 T^{2} )\)

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
\(3\) 3.80.2 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);