↑

Properties

Label 2432.a.19456.1
Conductor $2432$
Discriminant $-19456$
Mordell-Weil group \(\Z \oplus \Z/{4}\Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = x^6 + 2x^5 + 2x^2 + x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^6 + 2x^5z + 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 + 8x^5 + 9x^2 + 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, 0, 2, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, 0, 2, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, 9, 0, 0, 8, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2432\) \(=\) \( 2^{7} \cdot 19 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-19456\) \(=\) \( - 2^{10} \cdot 19 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(792\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 11 \)
\( I_6 \)  \(=\) \(22860\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 127 \)
\( I_{10} \)  \(=\) \(-2432\) \(=\)  \( - 2^{7} \cdot 19 \)
\( J_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(72\) \(=\)  \( 2^{3} \cdot 3^{2} \)
\( J_6 \)  \(=\) \(1280\) \(=\)  \( 2^{8} \cdot 5 \)
\( J_8 \)  \(=\) \(37104\) \(=\)  \( 2^{4} \cdot 3 \cdot 773 \)
\( J_{10} \)  \(=\) \(-19456\) \(=\)  \( - 2^{10} \cdot 19 \)
\( g_1 \)  \(=\) \(-24300000/19\)
\( g_2 \)  \(=\) \(-121500/19\)
\( g_3 \)  \(=\) \(-18000/19\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 2)\)
\((-2 : -2 : 1)\) \((-2 : 3 : 1)\) \((-1 : -3 : 2)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 2)\)
\((-2 : -2 : 1)\) \((-2 : 3 : 1)\) \((-1 : -3 : 2)\)
All points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -2 : 2)\)
\((-1 : 2 : 2)\) \((-2 : -5 : 1)\) \((-2 : 5 : 1)\)

magma: [C![-2,-2,1],C![-2,3,1],C![-1,-3,2],C![-1,-1,2],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // minimal model
 
magma: [C![-2,-5,1],C![-2,5,1],C![-1,-2,2],C![-1,2,2],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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: \(\Z \oplus \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0.059373\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0.059373\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -2 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 + xz^2 + 3z^3\) \(0.059373\) \(\infty\)
\((0 : -1 : 1) - (1 : -2 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 + xz^2 - z^3\) \(0\) \(4\)

2-torsion field: 6.2.184832.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.059373 \)
Real period: \( 23.72844 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.352211 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(10\) \(4\) \(1 + T\)
\(19\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 19 T^{2} )\)

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);