↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

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

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: X = HyperellipticCurve(R([0, -20, 73, -76, 4, 20]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3496\) \(=\)  \( 2^{3} \cdot 19 \cdot 23 \)
\( I_4 \)  \(=\) \(50536\) \(=\)  \( 2^{3} \cdot 6317 \)
\( I_6 \)  \(=\) \(55764955\) \(=\)  \( 5 \cdot 11152991 \)
\( I_{10} \)  \(=\) \(92500\) \(=\)  \( 2^{2} \cdot 5^{4} \cdot 37 \)
\( J_2 \)  \(=\) \(1748\) \(=\)  \( 2^{2} \cdot 19 \cdot 23 \)
\( J_4 \)  \(=\) \(118890\) \(=\)  \( 2 \cdot 3^{2} \cdot 5 \cdot 1321 \)
\( J_6 \)  \(=\) \(10257041\) \(=\)  \( 10257041 \)
\( J_8 \)  \(=\) \(948618892\) \(=\)  \( 2^{2} \cdot 13 \cdot 18242671 \)
\( J_{10} \)  \(=\) \(23125\) \(=\)  \( 5^{4} \cdot 37 \)
\( g_1 \)  \(=\) \(16319511005139968/23125\)
\( g_2 \)  \(=\) \(126998797147776/4625\)
\( g_3 \)  \(=\) \(31340429803664/23125\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -50 : 5)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -50 : 5)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (4 : 0 : 5)\)

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![4,-50,5]]; // minimal model
 
magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![4,0,5]]; // 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: \(\Z/{2}\Z \oplus \Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (4 : -50 : 5) - 2 \cdot(1 : 0 : 0)\) \(x (5x - 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\((1 : 0 : 1) + (4 : -50 : 5) - 2 \cdot(1 : 0 : 0)\) \((x - z) (5x - 4z)\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 - 2z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (4 : -50 : 5) - 2 \cdot(1 : 0 : 0)\) \(x (5x - 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\((1 : 0 : 1) + (4 : -50 : 5) - 2 \cdot(1 : 0 : 0)\) \((x - z) (5x - 4z)\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 - 2z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x (5x - 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\((1 : 1 : 1) + (4 : 0 : 5) - 2 \cdot(1 : 0 : 0)\) \((x - z) (5x - 4z)\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2 - 4z^3\) \(0\) \(8\)

2-torsion field: 3.3.148.1

BSD invariants

Hasse-Weil conjecture: unverified
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 Cluster picture
\(5\) \(2\) \(4\) \(4\) \(( 1 - T )( 1 + T )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 37 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.120.3 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);