Properties

Label 126859.b.126859.1
Conductor $126859$
Discriminant $-126859$
Mordell-Weil group \(\Z\)
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

$y^2 + y = x^5 - 8x^4 - 38x^3 - 29x^2 - 8x - 1$ (homogenize, simplify)
$y^2 + z^3y = x^5z - 8x^4z^2 - 38x^3z^3 - 29x^2z^4 - 8xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 32x^4 - 152x^3 - 116x^2 - 32x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -8, -29, -38, -8, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -8, -29, -38, -8, 1], R![1]);
 
sage: X = HyperellipticCurve(R([-3, -32, -116, -152, -32, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(126859\) \(=\) \( 126859 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-126859\) \(=\) \( -126859 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(18528\) \(=\)  \( 2^{5} \cdot 3 \cdot 193 \)
\( I_4 \)  \(=\) \(46416\) \(=\)  \( 2^{4} \cdot 3 \cdot 967 \)
\( I_6 \)  \(=\) \(283353360\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \cdot 367 \cdot 3217 \)
\( I_{10} \)  \(=\) \(-507436\) \(=\)  \( - 2^{2} \cdot 126859 \)
\( J_2 \)  \(=\) \(9264\) \(=\)  \( 2^{4} \cdot 3 \cdot 193 \)
\( J_4 \)  \(=\) \(3568168\) \(=\)  \( 2^{3} \cdot 577 \cdot 773 \)
\( J_6 \)  \(=\) \(1828822192\) \(=\)  \( 2^{4} \cdot 17 \cdot 79 \cdot 85109 \)
\( J_8 \)  \(=\) \(1052596477616\) \(=\)  \( 2^{4} \cdot 137 \cdot 197 \cdot 2437559 \)
\( J_{10} \)  \(=\) \(-126859\) \(=\)  \( -126859 \)
\( g_1 \)  \(=\) \(-68232727503987277824/126859\)
\( g_2 \)  \(=\) \(-2836879788910804992/126859\)
\( g_3 \)  \(=\) \(-156952622199877632/126859\)

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

magma: [C![1,0,0]]; // minimal model
 
magma: [C![1,0,0]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(1.302295\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(1.302295\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 + z^3\) \(1.302295\) \(\infty\)

2-torsion field: 5.3.2029744.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 1.302295 \)
Real period: \( 3.803713 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 4.953558 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(126859\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 551 T + 126859 T^{2} )\)

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
\(2\) 2.6.1

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 \(\Q\).