Properties

Label 1702.a.39146.1
Conductor $1702$
Discriminant $39146$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
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

$y^2 + (x + 1)y = x^5 - 13x^3 + 14x^2 + 22x - 30$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 13x^3z^3 + 14x^2z^4 + 22xz^5 - 30z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 52x^3 + 57x^2 + 90x - 119$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-30, 22, 14, -13, 0, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-30, 22, 14, -13, 0, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([-119, 90, 57, -52, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1702\) \(=\) \( 2 \cdot 23 \cdot 37 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(39146\) \(=\) \( 2 \cdot 23^{2} \cdot 37 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(7656\) \(=\)  \( 2^{3} \cdot 3 \cdot 11 \cdot 29 \)
\( I_4 \)  \(=\) \(14184\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 197 \)
\( I_6 \)  \(=\) \(36303939\) \(=\)  \( 3^{2} \cdot 7 \cdot 59 \cdot 9767 \)
\( I_{10} \)  \(=\) \(156584\) \(=\)  \( 2^{3} \cdot 23^{2} \cdot 37 \)
\( J_2 \)  \(=\) \(3828\) \(=\)  \( 2^{2} \cdot 3 \cdot 11 \cdot 29 \)
\( J_4 \)  \(=\) \(608202\) \(=\)  \( 2 \cdot 3^{3} \cdot 7 \cdot 1609 \)
\( J_6 \)  \(=\) \(128326985\) \(=\)  \( 5 \cdot 1607 \cdot 15971 \)
\( J_8 \)  \(=\) \(30331506444\) \(=\)  \( 2^{2} \cdot 3 \cdot 19 \cdot 283 \cdot 470081 \)
\( J_{10} \)  \(=\) \(39146\) \(=\)  \( 2 \cdot 23^{2} \cdot 37 \)
\( g_1 \)  \(=\) \(410988481022237184/19573\)
\( g_2 \)  \(=\) \(17058217029682752/19573\)
\( g_3 \)  \(=\) \(940225127082120/19573\)

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: \((1 : 0 : 0),\, (2 : -1 : 1),\, (2 : -2 : 1),\, (16 : -978 : 9),\, (16 : -1047 : 9)\)
All points: \((1 : 0 : 0),\, (2 : -1 : 1),\, (2 : -2 : 1),\, (16 : -978 : 9),\, (16 : -1047 : 9)\)
All points: \((1 : 0 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1),\, (16 : -69 : 9),\, (16 : 69 : 9)\)

magma: [C![1,0,0],C![2,-2,1],C![2,-1,1],C![16,-1047,9],C![16,-978,9]]; // minimal model
 
magma: [C![1,0,0],C![2,-1,1],C![2,1,1],C![16,-69,9],C![16,69,9]]; // 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: \(\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 7xz + 9z^2\) \(=\) \(0,\) \(y\) \(=\) \(-15xz^2 + 24z^3\) \(0.036854\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz - 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 7xz + 9z^2\) \(=\) \(0,\) \(y\) \(=\) \(-15xz^2 + 24z^3\) \(0.036854\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz - 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 7xz + 9z^2\) \(=\) \(0,\) \(y\) \(=\) \(-29xz^2 + 49z^3\) \(0.036854\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 2xz - 7z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 6.6.2803712.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.036854 \)
Real period: \( 15.53588 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.286283 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(23\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 4 T + 23 T^{2} )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 37 T^{2} )\)

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?
\(2\) 2.60.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

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);