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Properties

Label 44178.a.88356.1
Conductor $44178$
Discriminant $-88356$
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

$y^2 + (x^3 + x^2)y = -5x^4 + x^3 + 61x^2 - 9x - 220$ (homogenize, simplify)
$y^2 + (x^3 + x^2z)y = -5x^4z^2 + x^3z^3 + 61x^2z^4 - 9xz^5 - 220z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 - 19x^4 + 4x^3 + 244x^2 - 36x - 880$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-220, -9, 61, 1, -5]), R([0, 0, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-220, -9, 61, 1, -5], R![0, 0, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-880, -36, 244, 4, -19, 2, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(44178\) \(=\) \( 2 \cdot 3 \cdot 37 \cdot 199 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-88356\) \(=\) \( - 2^{2} \cdot 3 \cdot 37 \cdot 199 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(70648\) \(=\)  \( 2^{3} \cdot 8831 \)
\( I_4 \)  \(=\) \(-3402716\) \(=\)  \( - 2^{2} \cdot 850679 \)
\( I_6 \)  \(=\) \(-84272408433\) \(=\)  \( - 3^{2} \cdot 36899 \cdot 253763 \)
\( I_{10} \)  \(=\) \(-353424\) \(=\)  \( - 2^{4} \cdot 3 \cdot 37 \cdot 199 \)
\( J_2 \)  \(=\) \(35324\) \(=\)  \( 2^{2} \cdot 8831 \)
\( J_4 \)  \(=\) \(52558160\) \(=\)  \( 2^{4} \cdot 5 \cdot 656977 \)
\( J_6 \)  \(=\) \(105828428929\) \(=\)  \( 69029 \cdot 1533101 \)
\( J_8 \)  \(=\) \(243980810225599\) \(=\)  \( 35677 \cdot 6838602187 \)
\( J_{10} \)  \(=\) \(-88356\) \(=\)  \( - 2^{2} \cdot 3 \cdot 37 \cdot 199 \)
\( g_1 \)  \(=\) \(-13749578635451892006656/22089\)
\( g_2 \)  \(=\) \(-579148304999836936960/22089\)
\( g_3 \)  \(=\) \(-33012780912822492676/22089\)

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

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-7 : 121 : 2),\, (-7 : 124 : 2)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-7 : 121 : 2),\, (-7 : 124 : 2)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-7 : -3 : 2),\, (-7 : 3 : 2)\)

Copy content magma:[C![-7,121,2],C![-7,124,2],C![1,-1,0],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-7,-3,2],C![-7,3,2],C![1,-1,0],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-7 : 121 : 2) - (1 : 0 : 0)\) \(z (2x + 7z)\) \(=\) \(0,\) \(4y\) \(=\) \(-4x^3 - 111z^3\) \(0.087967\) \(\infty\)
Generator $D_0$ Height Order
\((-7 : 121 : 2) - (1 : 0 : 0)\) \(z (2x + 7z)\) \(=\) \(0,\) \(4y\) \(=\) \(-4x^3 - 111z^3\) \(0.087967\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(z (2x + 7z)\) \(=\) \(0,\) \(4y\) \(=\) \(-7x^3 + x^2z - 222z^3\) \(0.087967\) \(\infty\)

2-torsion field: 6.0.2759104381688064.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.087967 \)
Real period: \( 7.925699 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.394414 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(1\) \(2\) \(2\) \(-1^*\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\) yes
\(3\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 3 T^{2} )\) yes
\(37\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 10 T + 37 T^{2} )\) yes
\(199\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 199 T^{2} )\) yes

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?
\(2\) 2.20.2 no

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content 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 \(\Q\).

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

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

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