Properties

Label 12544.b.200704.1
Conductor $12544$
Discriminant $200704$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 7x^4 + 8x^3 + 15x^2 + 4x$ (homogenize, simplify)
$y^2 = x^5z - 7x^4z^2 + 8x^3z^3 + 15x^2z^4 + 4xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - 7x^4 + 8x^3 + 15x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 15, 8, -7, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 4, 15, 8, -7, 1], R![]);
 
sage: X = HyperellipticCurve(R([0, 4, 15, 8, -7, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(12544\) \(=\) \( 2^{8} \cdot 7^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(12544,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(200704\) \(=\) \( 2^{12} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2224\) \(=\)  \( 2^{4} \cdot 139 \)
\( I_4 \)  \(=\) \(9940\) \(=\)  \( 2^{2} \cdot 5 \cdot 7 \cdot 71 \)
\( I_6 \)  \(=\) \(7386820\) \(=\)  \( 2^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 2777 \)
\( I_{10} \)  \(=\) \(784\) \(=\)  \( 2^{4} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(4448\) \(=\)  \( 2^{5} \cdot 139 \)
\( J_4 \)  \(=\) \(797856\) \(=\)  \( 2^{5} \cdot 3 \cdot 8311 \)
\( J_6 \)  \(=\) \(183931136\) \(=\)  \( 2^{8} \cdot 743 \cdot 967 \)
\( J_8 \)  \(=\) \(45387874048\) \(=\)  \( 2^{8} \cdot 11 \cdot 17 \cdot 83 \cdot 11423 \)
\( J_{10} \)  \(=\) \(200704\) \(=\)  \( 2^{12} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(425073415774208/49\)
\( g_2 \)  \(=\) \(17141897862912/49\)
\( g_3 \)  \(=\) \(888433369664/49\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1)\)

magma: [C![0,0,1],C![1,0,0],C![4,0,1]]; // minimal model
 
magma: [C![0,0,1],C![1,0,0],C![4,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((4 : 0 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\((4 : 0 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(4\)
Generator $D_0$ Height Order
\((4 : 0 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-1/2xz^2\) \(0\) \(4\)

2-torsion field: \(\Q(\zeta_{7})^+\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 13.95491 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.872182 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(12\) \(4\) \(1\)
\(7\) \(2\) \(2\) \(1\) \(1 + 7 T^{2}\)

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
\(2\) 2.240.1

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 \(\Q\).