Properties

Label 195029.a.195029.1
Conductor $195029$
Discriminant $195029$
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

$y^2 + y = x^5 - 17x^4 + 2x^3 + 19x^2 - x - 6$ (homogenize, simplify)
$y^2 + z^3y = x^5z - 17x^4z^2 + 2x^3z^3 + 19x^2z^4 - xz^5 - 6z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 68x^4 + 8x^3 + 76x^2 - 4x - 23$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -1, 19, 2, -17, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -1, 19, 2, -17, 1], R![1]);
 
sage: X = HyperellipticCurve(R([-23, -4, 76, 8, -68, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(195029\) \(=\) \( 195029 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(195029\) \(=\) \( 195029 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(20608\) \(=\)  \( 2^{7} \cdot 7 \cdot 23 \)
\( I_4 \)  \(=\) \(27274864\) \(=\)  \( 2^{4} \cdot 1704679 \)
\( I_6 \)  \(=\) \(139843723584\) \(=\)  \( 2^{6} \cdot 3 \cdot 97 \cdot 7508791 \)
\( I_{10} \)  \(=\) \(780116\) \(=\)  \( 2^{2} \cdot 195029 \)
\( J_2 \)  \(=\) \(10304\) \(=\)  \( 2^{6} \cdot 7 \cdot 23 \)
\( J_4 \)  \(=\) \(-121960\) \(=\)  \( - 2^{3} \cdot 5 \cdot 3049 \)
\( J_6 \)  \(=\) \(5337536\) \(=\)  \( 2^{6} \cdot 83399 \)
\( J_8 \)  \(=\) \(10030932336\) \(=\)  \( 2^{4} \cdot 3 \cdot 67 \cdot 73 \cdot 42727 \)
\( J_{10} \)  \(=\) \(195029\) \(=\)  \( 195029 \)
\( g_1 \)  \(=\) \(116152684096230785024/195029\)
\( g_2 \)  \(=\) \(-133424310061629440/195029\)
\( g_3 \)  \(=\) \(566699092606976/195029\)

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: 5.1.3120464.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.386261 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.386261 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(195029\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 368 T + 195029 T^{2} )\)

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);