Properties

Label 676.b.17576.1
Conductor 676
Discriminant -17576
Mordell-Weil group \(\Z/{3}\Z \times \Z/{3}\Z\)
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = -x^6 + 3x^5 - 6x^4 + 6x^3 - 6x^2 + 3x - 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = -x^6 + 3x^5z - 6x^4z^2 + 6x^3z^3 - 6x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 + 12x^5 - 23x^4 + 26x^3 - 23x^2 + 12x - 4$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -6, 6, -6, 3, -1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -6, 6, -6, 3, -1]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-4, 12, -23, 26, -23, 12, -4]))
 

Invariants

Conductor: \( N \)  =  \(676\) = \( 2^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-17576\) = \( - 2^{3} \cdot 13^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-2488\) =  \( - 2^{3} \cdot 311 \)
\( I_4 \)  = \(4996\) =  \( 2^{2} \cdot 1249 \)
\( I_6 \)  = \(-1033336\) =  \( - 2^{3} \cdot 37 \cdot 3491 \)
\( I_{10} \)  = \(-71991296\) =  \( - 2^{15} \cdot 13^{3} \)
\( J_2 \)  = \(-311\) =  \( - 311 \)
\( J_4 \)  = \(3978\) =  \( 2 \cdot 3^{2} \cdot 13 \cdot 17 \)
\( J_6 \)  = \(-72332\) =  \( - 2^{2} \cdot 13^{2} \cdot 107 \)
\( J_8 \)  = \(1667692\) =  \( 2^{2} \cdot 13^{2} \cdot 2467 \)
\( J_{10} \)  = \(-17576\) =  \( - 2^{3} \cdot 13^{3} \)
\( g_1 \)  = \(2909390022551/17576\)
\( g_2 \)  = \(4602275343/676\)
\( g_3 \)  = \(10349147/26\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $D_6$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$, $\Q_{2}$, and $\Q_{11}$.

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: \(\Z/{3}\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 3.1.104.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 7.177120 \)
Tamagawa product: \( 3 \)
Torsion order:\( 9 \)
Leading coefficient: \( 0.265819 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(13\) \(3\) \(2\) \(3\) \(( 1 - T )^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_1$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 26.a2

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{}) \otimes \Q\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).