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Properties

Label 784.b.12544.1
Conductor $784$
Discriminant $12544$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(784\) \(=\) \( 2^{4} \cdot 7^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(12544\) \(=\) \( 2^{8} \cdot 7^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(116\) \(=\)  \( 2^{2} \cdot 29 \)
\( I_4 \)  \(=\) \(445\) \(=\)  \( 5 \cdot 89 \)
\( I_6 \)  \(=\) \(16259\) \(=\)  \( 71 \cdot 229 \)
\( I_{10} \)  \(=\) \(1568\) \(=\)  \( 2^{5} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(116\) \(=\)  \( 2^{2} \cdot 29 \)
\( J_4 \)  \(=\) \(264\) \(=\)  \( 2^{3} \cdot 3 \cdot 11 \)
\( J_6 \)  \(=\) \(-1280\) \(=\)  \( - 2^{8} \cdot 5 \)
\( J_8 \)  \(=\) \(-54544\) \(=\)  \( - 2^{4} \cdot 7 \cdot 487 \)
\( J_{10} \)  \(=\) \(12544\) \(=\)  \( 2^{8} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(82044596/49\)
\( g_2 \)  \(=\) \(1609674/49\)
\( g_3 \)  \(=\) \(-67280/49\)

  (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

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

Rational points

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

Copy content magma:[C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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

This curve is locally solvable everywhere.

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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - 2z^3\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 3xz^2\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{-7}) \)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 11.27010 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.313058 \)
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?
\(2\) \(4\) \(8\) \(4\) \(1^*\) \(1 + T\) no
\(7\) \(2\) \(2\) \(1\) \(1\) \(( 1 - T )^{2}\) 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?
\(2\) 2.360.1 yes
\(3\) 3.720.4 yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 56.b
  Elliptic curve isogeny class 14.a

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

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);