Properties

Label 784.a.1568.1
Conductor 784
Discriminant 1568
Mordell-Weil group \(\Z/{12}\Z\)
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

Related objects

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Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -2x^4 + 3x^2 - 2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -2x^4z^2 + 3x^2z^4 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 13x^2 - 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 0, 3, 0, -2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 0, 3, 0, -2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, 0, 13, 0, -6, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(784\) \(=\) \( 2^{4} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(1568\) \(=\) \( 2^{5} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3168\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 11 \)
\( I_4 \)  \(=\) \(1920\) \(=\)  \( 2^{7} \cdot 3 \cdot 5 \)
\( I_6 \)  \(=\) \(974592\) \(=\)  \( 2^{8} \cdot 3^{4} \cdot 47 \)
\( I_{10} \)  \(=\) \(6422528\) \(=\)  \( 2^{17} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(396\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 11 \)
\( J_4 \)  \(=\) \(6514\) \(=\)  \( 2 \cdot 3257 \)
\( J_6 \)  \(=\) \(144256\) \(=\)  \( 2^{7} \cdot 7^{2} \cdot 23 \)
\( J_8 \)  \(=\) \(3673295\) \(=\)  \( 5 \cdot 734659 \)
\( J_{10} \)  \(=\) \(1568\) \(=\)  \( 2^{5} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(304316815968/49\)
\( g_2 \)  \(=\) \(12641055372/49\)
\( g_3 \)  \(=\) \(14427072\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1)\)

magma: [C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(12\)

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 20.79335 \)
Tamagawa product: \( 2 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.288796 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(5\) \(2\) \(1 + T\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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 56.a4

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 \(\Q\).