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Properties

Label 8452.b.16904.1
Conductor $8452$
Discriminant $16904$
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

$y^2 + (x^3 + x^2 + 1)y = -3x^4 - 4x^3 + 10x^2 + 5x - 12$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + z^3)y = -3x^4z^2 - 4x^3z^3 + 10x^2z^4 + 5xz^5 - 12z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 - 11x^4 - 14x^3 + 42x^2 + 20x - 47$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-12, 5, 10, -4, -3]), R([1, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-12, 5, 10, -4, -3], R![1, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-47, 20, 42, -14, -11, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8452\) \(=\) \( 2^{2} \cdot 2113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(16904\) \(=\) \( 2^{3} \cdot 2113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(10724\) \(=\)  \( 2^{2} \cdot 7 \cdot 383 \)
\( I_4 \)  \(=\) \(47809\) \(=\)  \( 47809 \)
\( I_6 \)  \(=\) \(168584513\) \(=\)  \( 168584513 \)
\( I_{10} \)  \(=\) \(2163712\) \(=\)  \( 2^{10} \cdot 2113 \)
\( J_2 \)  \(=\) \(2681\) \(=\)  \( 7 \cdot 383 \)
\( J_4 \)  \(=\) \(297498\) \(=\)  \( 2 \cdot 3 \cdot 179 \cdot 277 \)
\( J_6 \)  \(=\) \(43749444\) \(=\)  \( 2^{2} \cdot 3 \cdot 59 \cdot 61 \cdot 1013 \)
\( J_8 \)  \(=\) \(7196799840\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \cdot 241 \cdot 62213 \)
\( J_{10} \)  \(=\) \(16904\) \(=\)  \( 2^{3} \cdot 2113 \)
\( g_1 \)  \(=\) \(138510937865757401/16904\)
\( g_2 \)  \(=\) \(2866450831711509/8452\)
\( g_3 \)  \(=\) \(78615136838721/4226\)

  (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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -6 : 1),\, (2 : -7 : 1),\, (-3 : 8 : 1),\, (-3 : 9 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -6 : 1),\, (2 : -7 : 1),\, (-3 : 8 : 1),\, (-3 : 9 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1),\, (-3 : -1 : 1),\, (-3 : 1 : 1)\)

magma: [C![-3,8,1],C![-3,9,1],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-6,1]]; // minimal model
 
magma: [C![-3,-1,1],C![-3,1,1],C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 3z^3\) \(0.011089\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 3z^3\) \(0.011089\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z - 2xz^2 - 5z^3\) \(0.011089\) \(\infty\)

2-torsion field: 6.6.1081856.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.011089 \)
Real period: \( 19.60059 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.652066 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(2113\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 62 T + 2113 T^{2} )\)

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?
\(2\) 2.10.1 no

Sato-Tate group

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

Decomposition of the Jacobian

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

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);