Properties

Label 3465.a.800415.1
Conductor $3465$
Discriminant $-800415$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 + (x^3 + 1)y = x^5 - 4x^3 - 3x^2 + 3x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 4x^3z^3 - 3x^2z^4 + 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 14x^3 - 12x^2 + 12x + 9$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 3, -3, -4, 0, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 3, -3, -4, 0, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, 12, -12, -14, 0, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3465\) \(=\) \( 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-800415\) \(=\) \( - 3^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(468\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 13 \)
\( I_4 \)  \(=\) \(-60903\) \(=\)  \( - 3^{2} \cdot 67 \cdot 101 \)
\( I_6 \)  \(=\) \(-7272387\) \(=\)  \( - 3^{2} \cdot 37 \cdot 21839 \)
\( I_{10} \)  \(=\) \(-102453120\) \(=\)  \( - 2^{7} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2} \)
\( J_2 \)  \(=\) \(117\) \(=\)  \( 3^{2} \cdot 13 \)
\( J_4 \)  \(=\) \(3108\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 37 \)
\( J_6 \)  \(=\) \(22240\) \(=\)  \( 2^{5} \cdot 5 \cdot 139 \)
\( J_8 \)  \(=\) \(-1764396\) \(=\)  \( - 2^{2} \cdot 3^{3} \cdot 17 \cdot 31^{2} \)
\( J_{10} \)  \(=\) \(-800415\) \(=\)  \( - 3^{3} \cdot 5 \cdot 7^{2} \cdot 11^{2} \)
\( g_1 \)  \(=\) \(-812017791/29645\)
\( g_2 \)  \(=\) \(-26337636/4235\)
\( g_3 \)  \(=\) \(-2255136/5929\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\) \((-3 : 13 : 1)\)
\((2 : 21 : 3)\) \((2 : -56 : 3)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\) \((-3 : 13 : 1)\)
\((2 : 21 : 3)\) \((2 : -56 : 3)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : 0 : 1)\) \((-3 : 0 : 1)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\)
\((2 : -77 : 3)\) \((2 : 77 : 3)\)

magma: [C![-3,13,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-56,3],C![2,21,3]]; // minimal model
 
magma: [C![-3,0,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-77,3],C![2,77,3]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + z^3\) \(0.079548\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 5z^3\) \(0\) \(2\)
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 3z)\) \(=\) \(0,\) \(2y\) \(=\) \(-7xz^2 + 5z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + z^3\) \(0.079548\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 5z^3\) \(0\) \(2\)
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 3z)\) \(=\) \(0,\) \(2y\) \(=\) \(-7xz^2 + 5z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 3 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 3z^3\) \(0.079548\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 3xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 6xz^2 - 9z^3\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + 3z)\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 14xz^2 + 11z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.079548 \)
Real period: \( 9.804870 \)
Tamagawa product: \( 8 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.389979 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(3\) \(2\) \(1 + 3 T^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 2 T + 7 T^{2} )\)
\(11\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 - 2 T + 11 T^{2} )\)

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?
\(2\) 2.180.3 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

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);