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Properties

Label 676.b.17576.1
Conductor $676$
Discriminant $-17576$
Mordell-Weil group \(\Z/{3}\Z \oplus \Z/{3}\Z\)
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-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)
$y^2 = -4x^6 + 12x^5 - 23x^4 + 26x^3 - 23x^2 + 12x - 4$ (homogenize, minimize)

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

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1244\) \(=\)  \( 2^{2} \cdot 311 \)
\( I_4 \)  \(=\) \(1249\) \(=\)  \( 1249 \)
\( I_6 \)  \(=\) \(129167\) \(=\)  \( 37 \cdot 3491 \)
\( I_{10} \)  \(=\) \(2249728\) \(=\)  \( 2^{10} \cdot 13^{3} \)
\( J_2 \)  \(=\) \(311\) \(=\)  \( 311 \)
\( J_4 \)  \(=\) \(3978\) \(=\)  \( 2 \cdot 3^{2} \cdot 13 \cdot 17 \)
\( J_6 \)  \(=\) \(72332\) \(=\)  \( 2^{2} \cdot 13^{2} \cdot 107 \)
\( J_8 \)  \(=\) \(1667692\) \(=\)  \( 2^{2} \cdot 13^{2} \cdot 2467 \)
\( J_{10} \)  \(=\) \(17576\) \(=\)  \( 2^{3} \cdot 13^{3} \)
\( g_1 \)  \(=\) \(2909390022551/17576\)
\( g_2 \)  \(=\) \(4602275343/676\)
\( g_3 \)  \(=\) \(10349147/26\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

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

Rational points

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

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

Copy content 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: \(\Z/{3}\Z \oplus \Z/{3}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

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

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 Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(3\) \(1\) \(1^*\) \(( 1 + T )^{2}\) yes
\(13\) \(2\) \(3\) \(3\) \(1\) \(( 1 - T )^{2}\) yes

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?
\(2\) 2.120.4 no
\(3\) 3.17280.1 yes

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 isogeny class:
  Elliptic curve isogeny class 26.a

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);