Properties

Label 588.a.18816.1
Conductor $588$
Discriminant $-18816$
Mordell-Weil group \(\Z/{24}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 + x^4 + 5x^2 + 12x + 8$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z + x^4z^2 + 5x^2z^4 + 12xz^5 + 8z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 4x^4 + 2x^3 + 20x^2 + 48x + 33$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 12, 5, 0, 1, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![8, 12, 5, 0, 1, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([33, 48, 20, 2, 4, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(588\) \(=\) \( 2^{2} \cdot 3 \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-18816\) \(=\) \( - 2^{7} \cdot 3 \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(748\) \(=\)  \( 2^{2} \cdot 11 \cdot 17 \)
\( I_4 \)  \(=\) \(11545\) \(=\)  \( 5 \cdot 2309 \)
\( I_6 \)  \(=\) \(2902787\) \(=\)  \( 2902787 \)
\( I_{10} \)  \(=\) \(2408448\) \(=\)  \( 2^{14} \cdot 3 \cdot 7^{2} \)
\( J_2 \)  \(=\) \(187\) \(=\)  \( 11 \cdot 17 \)
\( J_4 \)  \(=\) \(976\) \(=\)  \( 2^{4} \cdot 61 \)
\( J_6 \)  \(=\) \(-192\) \(=\)  \( - 2^{6} \cdot 3 \)
\( J_8 \)  \(=\) \(-247120\) \(=\)  \( - 2^{4} \cdot 5 \cdot 3089 \)
\( J_{10} \)  \(=\) \(18816\) \(=\)  \( 2^{7} \cdot 3 \cdot 7^{2} \)
\( g_1 \)  \(=\) \(228669389707/18816\)
\( g_2 \)  \(=\) \(398891383/1176\)
\( g_3 \)  \(=\) \(-34969/98\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-2 : 3 : 1),\, (-2 : 4 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-2 : 3 : 1),\, (-2 : 4 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : -1 : 1),\, (-2 : 1 : 1),\, (-1 : -2 : 1),\, (-1 : 2 : 1)\)

magma: [C![-2,3,1],C![-2,4,1],C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-2,-1,1],C![-2,1,1],C![-1,-2,1],C![-1,2,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z/{24}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 4 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x + z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2 - 6z^3\) \(0\) \(24\)
Generator $D_0$ Height Order
\((-2 : 4 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x + z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2 - 6z^3\) \(0\) \(24\)
Generator $D_0$ Height Order
\((-2 : 1 : 1) + (-1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x + z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 10xz^2 - 11z^3\) \(0\) \(24\)

2-torsion field: 8.0.12446784.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 20.65814 \)
Tamagawa product: \( 8 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.286918 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(7\) \(8\) \(( 1 - T )( 1 + T )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.45.1 yes
\(3\) 3.720.4 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 42.a

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);