Properties

Label 858491.a.858491.1
Conductor $858491$
Discriminant $-858491$
Mordell-Weil group \(\Z/{5}\Z\)
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

$y^2 + (x^2 + 1)y = x^5 - 8x^4 + 13x^3 + 83x - 212$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z - 8x^4z^2 + 13x^3z^3 + 83xz^5 - 212z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 31x^4 + 52x^3 + 2x^2 + 332x - 847$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-212, 83, 0, 13, -8, 1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-212, 83, 0, 13, -8, 1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-847, 332, 2, 52, -31, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(858491\) \(=\) \( 409 \cdot 2099 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-858491\) \(=\) \( - 409 \cdot 2099 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(17584\) \(=\)  \( 2^{4} \cdot 7 \cdot 157 \)
\( I_4 \)  \(=\) \(547708\) \(=\)  \( 2^{2} \cdot 7 \cdot 31 \cdot 631 \)
\( I_6 \)  \(=\) \(5545501647\) \(=\)  \( 3 \cdot 7^{2} \cdot 71 \cdot 531331 \)
\( I_{10} \)  \(=\) \(-3433964\) \(=\)  \( - 2^{2} \cdot 409 \cdot 2099 \)
\( J_2 \)  \(=\) \(8792\) \(=\)  \( 2^{3} \cdot 7 \cdot 157 \)
\( J_4 \)  \(=\) \(3129518\) \(=\)  \( 2 \cdot 7 \cdot 23 \cdot 9719 \)
\( J_6 \)  \(=\) \(1179953761\) \(=\)  \( 7^{2} \cdot 197 \cdot 251 \cdot 487 \)
\( J_8 \)  \(=\) \(145067638597\) \(=\)  \( 7^{2} \cdot 2960564053 \)
\( J_{10} \)  \(=\) \(-858491\) \(=\)  \( - 409 \cdot 2099 \)
\( g_1 \)  \(=\) \(-52533749281767391232/858491\)
\( g_2 \)  \(=\) \(-2126867779553219584/858491\)
\( g_3 \)  \(=\) \(-91209557279331904/858491\)

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: \((1 : 0 : 0),\, (4 : -8 : 1),\, (4 : -9 : 1)\)
All points: \((1 : 0 : 0),\, (4 : -8 : 1),\, (4 : -9 : 1)\)
All points: \((1 : 0 : 0),\, (4 : -1 : 1),\, (4 : 1 : 1)\)

magma: [C![1,0,0],C![4,-9,1],C![4,-8,1]]; // minimal model
 
magma: [C![1,0,0],C![4,-1,1],C![4,1,1]]; // 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: \(\Z/{5}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((4 : -8 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(-8z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\((4 : -8 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(-8z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\((4 : 1 : 1) - (1 : 0 : 0)\) \(x - 4z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 15z^3\) \(0\) \(5\)

2-torsion field: 5.3.3433964.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 5.049862 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
Leading coefficient: \( 1.817950 \)
Analytic order of Ш: \( 9 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(409\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 25 T + 409 T^{2} )\)
\(2099\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 60 T + 2099 T^{2} )\)

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?
\(2\) 2.6.1 no
\(5\) not computed 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);