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Properties

Label 471900.a.943800.1
Conductor $471900$
Discriminant $943800$
Mordell-Weil group \(\Z/{2}\Z\)
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

$y^2 + (x^3 + 1)y = x^6 - 31x^5 - 56x^4 + 7x^3 + 20x^2 - 3x - 1$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^6 - 31x^5z - 56x^4z^2 + 7x^3z^3 + 20x^2z^4 - 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 5x^6 - 124x^5 - 224x^4 + 30x^3 + 80x^2 - 12x - 3$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -3, 20, 7, -56, -31, 1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -3, 20, 7, -56, -31, 1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-3, -12, 80, 30, -224, -124, 5]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(471900\) \(=\) \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(943800\) \(=\) \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(177620\) \(=\)  \( 2^{2} \cdot 5 \cdot 83 \cdot 107 \)
\( I_4 \)  \(=\) \(621435529\) \(=\)  \( 11 \cdot 13 \cdot 4345703 \)
\( I_6 \)  \(=\) \(35388246892229\) \(=\)  \( 7 \cdot 11^{2} \cdot 73 \cdot 572338259 \)
\( I_{10} \)  \(=\) \(120806400\) \(=\)  \( 2^{10} \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13 \)
\( J_2 \)  \(=\) \(44405\) \(=\)  \( 5 \cdot 83 \cdot 107 \)
\( J_4 \)  \(=\) \(56265354\) \(=\)  \( 2 \cdot 3^{4} \cdot 347317 \)
\( J_6 \)  \(=\) \(30561804868\) \(=\)  \( 2^{2} \cdot 7 \cdot 1091493031 \)
\( J_8 \)  \(=\) \(-452173278895444\) \(=\)  \( - 2^{2} \cdot 19 \cdot 61 \cdot 15017 \cdot 6494987 \)
\( J_{10} \)  \(=\) \(943800\) \(=\)  \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13 \)
\( g_1 \)  \(=\) \(6905885338921613550125/37752\)
\( g_2 \)  \(=\) \(32843196581350130595/6292\)
\( g_3 \)  \(=\) \(602618898499869937/9438\)

  (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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\Q_{2}$.

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: \(\Z/{2}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(50y\) \(=\) \(-51xz^2 - 7z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(50y\) \(=\) \(-51xz^2 - 7z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(5x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(50y\) \(=\) \(x^3 - 102xz^2 - 13z^3\) \(0\) \(2\)

2-torsion field: 6.6.13361376600.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 1.063965 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.127931 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(3\) \(1\) \(-1^*\) \(( 1 - T )( 1 + T )\) yes
\(3\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 3 T^{2} )\) yes
\(5\) \(2\) \(2\) \(1\) \(-1\) \(( 1 - T )( 1 + T )\) yes
\(11\) \(2\) \(2\) \(1\) \(-1\) \(1 - 4 T + 11 T^{2}\) yes
\(13\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 2 T + 13 T^{2} )\) yes

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?
\(2\) 2.15.1 yes
\(3\) 3.40.1 no

Sato-Tate group

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

Decomposition of the Jacobian

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

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);