Properties

Label 111989.a.111989.2
Conductor $111989$
Discriminant $111989$
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

$y^2 + xy = x^5 + 5x^4 - 8x^3 - 48x^2 + x$ (homogenize, simplify)
$y^2 + xz^2y = x^5z + 5x^4z^2 - 8x^3z^3 - 48x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 20x^4 - 32x^3 - 191x^2 + 4x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -48, -8, 5, 1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -48, -8, 5, 1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, -191, -32, 20, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(111989\) \(=\) \( 53 \cdot 2113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(111989\) \(=\) \( 53 \cdot 2113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(16976\) \(=\)  \( 2^{4} \cdot 1061 \)
\( I_4 \)  \(=\) \(7208980\) \(=\)  \( 2^{2} \cdot 5 \cdot 19 \cdot 61 \cdot 311 \)
\( I_6 \)  \(=\) \(34342392335\) \(=\)  \( 5 \cdot 16603 \cdot 413689 \)
\( I_{10} \)  \(=\) \(447956\) \(=\)  \( 2^{2} \cdot 53 \cdot 2113 \)
\( J_2 \)  \(=\) \(8488\) \(=\)  \( 2^{3} \cdot 1061 \)
\( J_4 \)  \(=\) \(1800426\) \(=\)  \( 2 \cdot 3 \cdot 101 \cdot 2971 \)
\( J_6 \)  \(=\) \(432614081\) \(=\)  \( 432614081 \)
\( J_8 \)  \(=\) \(107623634513\) \(=\)  \( 107623634513 \)
\( J_{10} \)  \(=\) \(111989\) \(=\)  \( 53 \cdot 2113 \)
\( g_1 \)  \(=\) \(44058210592401031168/111989\)
\( g_2 \)  \(=\) \(1101010317277135872/111989\)
\( g_3 \)  \(=\) \(31168176376153664/111989\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 0 : 1)\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 3.3.447956.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)   (lower bound)
2-Selmer rank:\(4\)
Regulator: \( 1.0 \)
Real period: \( 2.154971 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 2.154971 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(53\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 53 T^{2} )\)
\(2113\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 14 T + 2113 T^{2} )\)

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);