Properties

Label 925.a.23125.1
Conductor 925
Discriminant 23125
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 18, -19, 1, 5], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -5, 18, -19, 1, 5]), R([0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 18, -19, 1, 5], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -20, 73, -76, 4, 20]))
 

$y^2 + xy = 5x^5 + x^4 - 19x^3 + 18x^2 - 5x$ (homogenize, simplify)
$y^2 + xz^2y = 5x^5z + x^4z^2 - 19x^3z^3 + 18x^2z^4 - 5xz^5$ (dehomogenize, simplify)
$y^2 = 20x^5 + 4x^4 - 76x^3 + 73x^2 - 20x$ (minimize, homogenize)

Invariants

\( N \)  =  \(925\) = \( 5^{2} \cdot 37 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(23125\) = \( 5^{4} \cdot 37 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(13984\) =  \( 2^{5} \cdot 19 \cdot 23 \)
\( I_4 \)  = \(808576\) =  \( 2^{7} \cdot 6317 \)
\( I_6 \)  = \(3568957120\) =  \( 2^{6} \cdot 5 \cdot 11152991 \)
\( I_{10} \)  = \(94720000\) =  \( 2^{12} \cdot 5^{4} \cdot 37 \)
\( J_2 \)  = \(1748\) =  \( 2^{2} \cdot 19 \cdot 23 \)
\( J_4 \)  = \(118890\) =  \( 2 \cdot 3^{2} \cdot 5 \cdot 1321 \)
\( J_6 \)  = \(10257041\) =  \( 10257041 \)
\( J_8 \)  = \(948618892\) =  \( 2^{2} \cdot 13 \cdot 18242671 \)
\( J_{10} \)  = \(23125\) =  \( 5^{4} \cdot 37 \)
\( g_1 \)  = \(16319511005139968/23125\)
\( g_2 \)  = \(126998797147776/4625\)
\( g_3 \)  = \(31340429803664/23125\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![4,-50,5]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -50 : 5)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(3\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{2}\Z \times \Z/{8}\Z\)

Generator Height Order
\((-5x + 4z) x\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\((x - z) (5x - 4z)\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 - 2z^3\) \(0\) \(8\)

2-torsion field: 3.3.148.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 20.87893 \)
Tamagawa product: \( 4 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.326233 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(4\) \(2\) \(4\) \(( 1 - T )( 1 + T )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 37 T^{2} )\)

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 \(\Q\).