Properties

Label 2304.a.13824.1
Conductor 2304
Discriminant 13824
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(2304\) \(=\) \( 2^{8} \cdot 3^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(13824\) \(=\) \( 2^{9} \cdot 3^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1632\) \(=\)  \( 2^{5} \cdot 3 \cdot 17 \)
\( I_4 \)  \(=\) \(59904\) \(=\)  \( 2^{9} \cdot 3^{2} \cdot 13 \)
\( I_6 \)  \(=\) \(29196288\) \(=\)  \( 2^{15} \cdot 3^{4} \cdot 11 \)
\( I_{10} \)  \(=\) \(56623104\) \(=\)  \( 2^{21} \cdot 3^{3} \)
\( J_2 \)  \(=\) \(204\) \(=\)  \( 2^{2} \cdot 3 \cdot 17 \)
\( J_4 \)  \(=\) \(1110\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 37 \)
\( J_6 \)  \(=\) \(4324\) \(=\)  \( 2^{2} \cdot 23 \cdot 47 \)
\( J_8 \)  \(=\) \(-87501\) \(=\)  \( - 3 \cdot 29167 \)
\( J_{10} \)  \(=\) \(13824\) \(=\)  \( 2^{9} \cdot 3^{3} \)
\( g_1 \)  \(=\) \(25557426\)
\( g_2 \)  \(=\) \(2726715/4\)
\( g_3 \)  \(=\) \(312409/24\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : 1 : 1)\)
\((1 : -2 : 1)\) \((-1 : -4 : 2)\) \((-4 : -249 : 9)\) \((-4 : -480 : 9)\)

magma: [C![-4,-480,9],C![-4,-249,9],C![-1,-4,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.049601\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 + 2xz - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)

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

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.049601 \)
Real period: \( 23.74493 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.294447 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(9\) \(2\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(3\) \(2\) \(1 + 2 T + 3 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

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

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 \(\Q\).