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Properties

Label 10040.c.502000.1
Conductor $10040$
Discriminant $-502000$
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

$y^2 + (x^2 + 1)y = x^5 - 3x^4 + 3x^3 - 2x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z - 3x^4z^2 + 3x^3z^3 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 11x^4 + 12x^3 + 2x^2 - 8x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 0, 3, -3, 1]), R([1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 0, 3, -3, 1], R![1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, -8, 2, 12, -11, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10040\) \(=\) \( 2^{3} \cdot 5 \cdot 251 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-502000\) \(=\) \( - 2^{4} \cdot 5^{3} \cdot 251 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(16\) \(=\)  \( 2^{4} \)
\( I_4 \)  \(=\) \(-4076\) \(=\)  \( - 2^{2} \cdot 1019 \)
\( I_6 \)  \(=\) \(38588\) \(=\)  \( 2^{2} \cdot 11 \cdot 877 \)
\( I_{10} \)  \(=\) \(2008000\) \(=\)  \( 2^{6} \cdot 5^{3} \cdot 251 \)
\( J_2 \)  \(=\) \(8\) \(=\)  \( 2^{3} \)
\( J_4 \)  \(=\) \(682\) \(=\)  \( 2 \cdot 11 \cdot 31 \)
\( J_6 \)  \(=\) \(-5796\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
\( J_8 \)  \(=\) \(-127873\) \(=\)  \( -127873 \)
\( J_{10} \)  \(=\) \(502000\) \(=\)  \( 2^{4} \cdot 5^{3} \cdot 251 \)
\( g_1 \)  \(=\) \(2048/31375\)
\( g_2 \)  \(=\) \(21824/31375\)
\( g_3 \)  \(=\) \(-23184/31375\)

  (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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (3 : 5 : 1),\, (3 : -15 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (3 : 5 : 1),\, (3 : -15 : 1)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (3 : -20 : 1),\, (3 : 20 : 1)\)

Copy content magma:[C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![3,-15,1],C![3,5,1]]; // minimal model
 
Copy content magma:[C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0],C![3,-20,1],C![3,20,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

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

Generator $D_0$ Height Order
\((0 : 0 : 1) + (3 : -15 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 3z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2\) \(0.045171\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (3 : -15 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 3z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2\) \(0.045171\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 1 : 1) + (3 : -20 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 3z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 10xz^2 + z^3\) \(0.045171\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - z^3\) \(0\) \(2\)

2-torsion field: 4.2.1255.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.045171 \)
Real period: \( 11.06047 \)
Tamagawa product: \( 6 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.749431 \)
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?
\(2\) \(3\) \(4\) \(2\) \(-1^*\) \(1 + T + 2 T^{2}\) no
\(5\) \(1\) \(3\) \(3\) \(-1\) \(( 1 - T )( 1 + 4 T + 5 T^{2} )\) yes
\(251\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 - 6 T + 251 T^{2} )\) 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?
\(2\) 2.30.3 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);