Properties

Label 110880.a.887040.1
Conductor $110880$
Discriminant $-887040$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 + (x^3 + x)y = -4x^6 - 45x^4 - 176x^2 - 231$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -4x^6 - 45x^4z^2 - 176x^2z^4 - 231z^6$ (dehomogenize, simplify)
$y^2 = -15x^6 - 178x^4 - 703x^2 - 924$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-231, 0, -176, 0, -45, 0, -4]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-231, 0, -176, 0, -45, 0, -4], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-924, 0, -703, 0, -178, 0, -15]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(110880\) \(=\) \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-887040\) \(=\) \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(666068\) \(=\)  \( 2^{2} \cdot 13 \cdot 12809 \)
\( I_4 \)  \(=\) \(38797\) \(=\)  \( 11 \cdot 3527 \)
\( I_6 \)  \(=\) \(8613261963\) \(=\)  \( 3^{2} \cdot 19 \cdot 2903 \cdot 17351 \)
\( I_{10} \)  \(=\) \(110880\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
\( J_2 \)  \(=\) \(666068\) \(=\)  \( 2^{2} \cdot 13 \cdot 12809 \)
\( J_4 \)  \(=\) \(18485248328\) \(=\)  \( 2^{3} \cdot 43 \cdot 6869 \cdot 7823 \)
\( J_6 \)  \(=\) \(684022107605760\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 24091 \cdot 32009 \)
\( J_8 \)  \(=\) \(28475207855231639024\) \(=\)  \( 2^{4} \cdot 826663 \cdot 2152873046153 \)
\( J_{10} \)  \(=\) \(887040\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
\( g_1 \)  \(=\) \(512097765844517969700609428/3465\)
\( g_2 \)  \(=\) \(21337377054806526458909866/3465\)
\( g_3 \)  \(=\) \(342108663769973403056\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 + 2z^3\) \(0.891621\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(1.006808\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(3xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 11z^2\) \(=\) \(0,\) \(3y\) \(=\) \(4xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 + 2z^3\) \(0.891621\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(1.006808\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(3xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 11z^2\) \(=\) \(0,\) \(3y\) \(=\) \(4xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 5xz^2 + 4z^3\) \(0.891621\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 5xz^2\) \(1.006808\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 + 7xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(3x^2 + 11z^2\) \(=\) \(0,\) \(3y\) \(=\) \(x^3 + 9xz^2\) \(0\) \(2\)

2-torsion field: 8.0.455583411360000.109

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(5\)
Regulator: \( 0.897691 \)
Real period: \( 3.441709 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.544796 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(8\) \(4\) \(1 + T\)
\(3\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 11 T^{2} )\)

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 240.a
  Elliptic curve isogeny class 462.a

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);