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Properties

Label 373248.e.746496.1
Conductor $373248$
Discriminant $746496$
Mordell-Weil group \(\Z/{3}\Z\)
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

$y^2 + (x^3 + x)y = x^4 - x^3 + 2x^2 - 3x - 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = x^4z^2 - x^3z^3 + 2x^2z^4 - 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 6x^4 - 4x^3 + 9x^2 - 12x - 4$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -3, 2, -1, 1]), R([0, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -3, 2, -1, 1], R![0, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-4, -12, 9, -4, 6, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(373248\) \(=\) \( 2^{9} \cdot 3^{6} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(746496\) \(=\) \( 2^{10} \cdot 3^{6} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8\) \(=\)  \( 2^{3} \)
\( I_4 \)  \(=\) \(486\) \(=\)  \( 2 \cdot 3^{5} \)
\( I_6 \)  \(=\) \(5184\) \(=\)  \( 2^{6} \cdot 3^{4} \)
\( I_{10} \)  \(=\) \(384\) \(=\)  \( 2^{7} \cdot 3 \)
\( J_2 \)  \(=\) \(24\) \(=\)  \( 2^{3} \cdot 3 \)
\( J_4 \)  \(=\) \(-2892\) \(=\)  \( - 2^{2} \cdot 3 \cdot 241 \)
\( J_6 \)  \(=\) \(-104944\) \(=\)  \( - 2^{4} \cdot 7 \cdot 937 \)
\( J_8 \)  \(=\) \(-2720580\) \(=\)  \( - 2^{2} \cdot 3 \cdot 5 \cdot 45343 \)
\( J_{10} \)  \(=\) \(746496\) \(=\)  \( 2^{10} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(32/3\)
\( g_2 \)  \(=\) \(-482/9\)
\( g_3 \)  \(=\) \(-6559/81\)

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

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

Automorphism group

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

Rational points

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

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

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{3}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.2.11943936.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 8.119701 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.706567 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(9\) \(10\) \(3\) \(-1^*\) \(1 - T\) no
\(3\) \(6\) \(6\) \(1\) \(-1^*\) \(1\) no

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?
\(2\) 2.20.1 no
\(3\) 3.240.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content 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 \(\Q\).

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

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

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