Properties

Label 2980.a.381440.1
Conductor $2980$
Discriminant $-381440$
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

$y^2 + (x^3 + 1)y = 2x^5 + 3x^4 - x^2 - x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 2x^5z + 3x^4z^2 - x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 12x^4 + 2x^3 - 4x^2 - 4x + 1$ (homogenize, minimize)

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

Invariants

Conductor: \( N \)  \(=\)  \(2980\) \(=\) \( 2^{2} \cdot 5 \cdot 149 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-381440\) \(=\) \( - 2^{9} \cdot 5 \cdot 149 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(364\) \(=\)  \( 2^{2} \cdot 7 \cdot 13 \)
\( I_4 \)  \(=\) \(1081\) \(=\)  \( 23 \cdot 47 \)
\( I_6 \)  \(=\) \(1867603\) \(=\)  \( 17 \cdot 109859 \)
\( I_{10} \)  \(=\) \(48824320\) \(=\)  \( 2^{16} \cdot 5 \cdot 149 \)
\( J_2 \)  \(=\) \(91\) \(=\)  \( 7 \cdot 13 \)
\( J_4 \)  \(=\) \(300\) \(=\)  \( 2^{2} \cdot 3 \cdot 5^{2} \)
\( J_6 \)  \(=\) \(-23056\) \(=\)  \( - 2^{4} \cdot 11 \cdot 131 \)
\( J_8 \)  \(=\) \(-547024\) \(=\)  \( - 2^{4} \cdot 179 \cdot 191 \)
\( J_{10} \)  \(=\) \(381440\) \(=\)  \( 2^{9} \cdot 5 \cdot 149 \)
\( g_1 \)  \(=\) \(6240321451/381440\)
\( g_2 \)  \(=\) \(11303565/19072\)
\( g_3 \)  \(=\) \(-11932921/23840\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\) \((-3 : 3 : 2)\) \((-3 : 16 : 2)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\) \((-3 : 3 : 2)\) \((-3 : 16 : 2)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((1 : -4 : 1)\) \((1 : 4 : 1)\) \((-3 : -13 : 2)\) \((-3 : 13 : 2)\)

magma: [C![-3,3,2],C![-3,16,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1]]; // minimal model
 
magma: [C![-3,-13,2],C![-3,13,2],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.010301\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.010301\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0.010301\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 6.2.177608000.4

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.010301 \)
Real period: \( 14.54382 \)
Tamagawa product: \( 12 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.449451 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(9\) \(12\) \(( 1 - T )( 1 + T )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(149\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 149 T^{2} )\)

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?
\(2\) 2.15.1 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);