Properties

Label 588.a.18816.1
Conductor 588
Discriminant -18816
Mordell-Weil group \(\Z/{24}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-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)
$y^2 = x^6 + 4x^5 + 4x^4 + 2x^3 + 20x^2 + 48x + 33$ (minimize, homogenize)

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

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

\( I_2 \)  = \(-1496\) =  \( - 2^{3} \cdot 11 \cdot 17 \)
\( I_4 \)  = \(46180\) =  \( 2^{2} \cdot 5 \cdot 2309 \)
\( I_6 \)  = \(-23222296\) =  \( - 2^{3} \cdot 2902787 \)
\( I_{10} \)  = \(-77070336\) =  \( - 2^{19} \cdot 3 \cdot 7^{2} \)
\( J_2 \)  = \(-187\) =  \( - 11 \cdot 17 \)
\( J_4 \)  = \(976\) =  \( 2^{4} \cdot 61 \)
\( J_6 \)  = \(192\) =  \( 2^{6} \cdot 3 \)
\( J_8 \)  = \(-247120\) =  \( - 2^{4} \cdot 5 \cdot 3089 \)
\( J_{10} \)  = \(-18816\) =  \( - 2^{7} \cdot 3 \cdot 7^{2} \)
\( g_1 \)  = \(228669389707/18816\)
\( g_2 \)  = \(398891383/1176\)
\( g_3 \)  = \(-34969/98\)

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

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

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]];
 

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
\((-2 : 4 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x + z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2 - 6z^3\) \(0\) \(24\)

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
\(2\) \(7\) \(2\) \(8\) \(( 1 - T )( 1 + T )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 14.a5
  Elliptic curve 42.a5

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 \(\Q\).