Properties

Label 76608.a.76608.1
Conductor $76608$
Discriminant $-76608$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 + xy = -21x^6 - 88x^4 - 123x^2 - 57$ (homogenize, simplify)
$y^2 + xz^2y = -21x^6 - 88x^4z^2 - 123x^2z^4 - 57z^6$ (dehomogenize, simplify)
$y^2 = -84x^6 - 352x^4 - 491x^2 - 228$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-57, 0, -123, 0, -88, 0, -21]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-57, 0, -123, 0, -88, 0, -21], R![0, 1]);
 
sage: X = HyperellipticCurve(R([-228, 0, -491, 0, -352, 0, -84]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(76608\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-76608\) \(=\) \( - 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(920224\) \(=\)  \( 2^{5} \cdot 149 \cdot 193 \)
\( I_4 \)  \(=\) \(53821\) \(=\)  \( 107 \cdot 503 \)
\( I_6 \)  \(=\) \(16505652732\) \(=\)  \( 2^{2} \cdot 3 \cdot 9437 \cdot 145753 \)
\( I_{10} \)  \(=\) \(9576\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
\( J_2 \)  \(=\) \(920224\) \(=\)  \( 2^{5} \cdot 149 \cdot 193 \)
\( J_4 \)  \(=\) \(35283806210\) \(=\)  \( 2 \cdot 5 \cdot 1039 \cdot 3395939 \)
\( J_6 \)  \(=\) \(1803829961380608\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 7 \cdot 19 \cdot 5886558719 \)
\( J_8 \)  \(=\) \(103745160429168513023\) \(=\)  \( 1069 \cdot 121291 \cdot 755147 \cdot 1059571 \)
\( J_{10} \)  \(=\) \(76608\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \)
\( g_1 \)  \(=\) \(10310691783200514787538108416/1197\)
\( g_2 \)  \(=\) \(429611720754327142357775360/1197\)
\( g_3 \)  \(=\) \(19939239196668773298176\)

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

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

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

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$.

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(14x^2 + 19z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(14x^2 + 19z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(14x^2 + 19z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)

2-torsion field: 8.0.1661007193767936.315

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(5\)
Regulator: \( 1 \)
Real period: \( 3.786797 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.893398 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 - T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 7 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 19 T^{2} )\)

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?
\(2\) 2.90.6 yes
\(3\) 3.90.1 no

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 57.c
  Elliptic curve isogeny class 1344.h

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 \(\Q\).

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

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

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