Properties

Label 980.a.878080.1
Conductor 980
Discriminant -878080
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, -4, 2, -4, 1, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, -4, 2, -4, 1, -1]), R([1, 0, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, -4, 2, -4, 1, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 4, -16, 10, -16, 4, -3]))
 

$y^2 + (x^3 + 1)y = -x^6 + x^5 - 4x^4 + 2x^3 - 4x^2 + x - 1$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^6 + x^5z - 4x^4z^2 + 2x^3z^3 - 4x^2z^4 + xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 4x^5 - 16x^4 + 10x^3 - 16x^2 + 4x - 3$ (minimize, homogenize)

Invariants

\( N \)  =  \(980\) = \( 2^{2} \cdot 5 \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-878080\) = \( - 2^{9} \cdot 5 \cdot 7^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(-5016\) =  \( - 2^{3} \cdot 3 \cdot 11 \cdot 19 \)
\( I_4 \)  = \(202980\) =  \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 199 \)
\( I_6 \)  = \(-333605784\) =  \( - 2^{3} \cdot 3 \cdot 2129 \cdot 6529 \)
\( I_{10} \)  = \(-3596615680\) =  \( - 2^{21} \cdot 5 \cdot 7^{3} \)
\( J_2 \)  = \(-627\) =  \( - 3 \cdot 11 \cdot 19 \)
\( J_4 \)  = \(14266\) =  \( 2 \cdot 7 \cdot 1019 \)
\( J_6 \)  = \(-359660\) =  \( - 2^{2} \cdot 5 \cdot 7^{2} \cdot 367 \)
\( J_8 \)  = \(5497016\) =  \( 2^{3} \cdot 7^{2} \cdot 37 \cdot 379 \)
\( J_{10} \)  = \(-878080\) =  \( - 2^{9} \cdot 5 \cdot 7^{3} \)
\( g_1 \)  = \(96903107471907/878080\)
\( g_2 \)  = \(251175228777/62720\)
\( g_3 \)  = \(144278343/896\)

Automorphism group

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

Rational points

magma: [];
 

This curve has no rational points.

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable except over $\R$ and $\Q_{2}$.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

Generator Height Order
\(3x^2 - xz + z^2\) \(=\) \(0,\) \(9y\) \(=\) \(4xz^2 - 7z^3\) \(0\) \(12\)

2-torsion field: 8.0.96040000.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 4.677173 \)
Tamagawa product: \( 12 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.389764 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(9\) \(2\) \(4\) \(( 1 - T )( 1 + T )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 5 T^{2} )\)
\(7\) \(3\) \(2\) \(3\) \(( 1 - T )( 1 + T )\)

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.a6
  Elliptic curve 70.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 \(\Q\).