Properties

Label 4312.a.189728.1
Conductor $4312$
Discriminant $189728$
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 + (x + 1)y = -2x^5 + 4x^4 - 2x^3 - 2x^2$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -2x^5z + 4x^4z^2 - 2x^3z^3 - 2x^2z^4$ (dehomogenize, simplify)
$y^2 = -8x^5 + 16x^4 - 8x^3 - 7x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, -2, 4, -2]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, -2, 4, -2], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -7, -8, 16, -8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(4312\) \(=\) \( 2^{3} \cdot 7^{2} \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(189728\) \(=\) \( 2^{5} \cdot 7^{2} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(384\) \(=\)  \( 2^{7} \cdot 3 \)
\( I_4 \)  \(=\) \(-6912\) \(=\)  \( - 2^{8} \cdot 3^{3} \)
\( I_6 \)  \(=\) \(-404100\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 449 \)
\( I_{10} \)  \(=\) \(758912\) \(=\)  \( 2^{7} \cdot 7^{2} \cdot 11^{2} \)
\( J_2 \)  \(=\) \(192\) \(=\)  \( 2^{6} \cdot 3 \)
\( J_4 \)  \(=\) \(2688\) \(=\)  \( 2^{7} \cdot 3 \cdot 7 \)
\( J_6 \)  \(=\) \(-156\) \(=\)  \( - 2^{2} \cdot 3 \cdot 13 \)
\( J_8 \)  \(=\) \(-1813824\) \(=\)  \( - 2^{6} \cdot 3^{2} \cdot 47 \cdot 67 \)
\( J_{10} \)  \(=\) \(189728\) \(=\)  \( 2^{5} \cdot 7^{2} \cdot 11^{2} \)
\( g_1 \)  \(=\) \(8153726976/5929\)
\( g_2 \)  \(=\) \(84934656/847\)
\( g_3 \)  \(=\) \(-179712/5929\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -6 : 2)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -6 : 2)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 2)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,0,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,0,2],C![1,0,0]]; // 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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -6 : 2) - 2 \cdot(1 : 0 : 0)\) \(x (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-3xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -6 : 2) - 2 \cdot(1 : 0 : 0)\) \(x (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-3xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2 + z^3\) \(0\) \(10\)

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 10.28259 \)
Tamagawa product: \( 10 \)
Torsion order:\( 10 \)
Leading coefficient: \( 1.028259 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.90.3

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 \(\Q\).