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Properties

Label 129600.b.129600.1
Conductor 129600129600
Discriminant 129600129600
Mordell-Weil group trivial
Sato-Tate group USp(4)\mathrm{USp}(4)
End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R R\R
End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q Q\Q
End⁡(J)⊗Q\End(J) \otimes \Q Q\Q
Q‾\overline{\Q}-simple yes
GL2\mathrm{GL}_2-type no

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Minimal equation

Minimal equation

Simplified equation

y2+xy=−x6−2x5−2x4−x3+x2+2x−1y^2 + xy = -x^6 - 2x^5 - 2x^4 - x^3 + x^2 + 2x - 1 (homogenize, simplify)
y2+xz2y=−x6−2x5z−2x4z2−x3z3+x2z4+2xz5−z6y^2 + xz^2y = -x^6 - 2x^5z - 2x^4z^2 - x^3z^3 + x^2z^4 + 2xz^5 - z^6 (dehomogenize, simplify)
y2=−4x6−8x5−8x4−4x3+5x2+8x−4y^2 = -4x^6 - 8x^5 - 8x^4 - 4x^3 + 5x^2 + 8x - 4 (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 2, 1, -1, -2, -2, -1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, 1, -1, -2, -2, -1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([-4, 8, 5, -4, -8, -8, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  129600129600 == 26⋅34⋅52 2^{6} \cdot 3^{4} \cdot 5^{2}
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  129600129600 == 26⋅34⋅52 2^{6} \cdot 3^{4} \cdot 5^{2}
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 708708 ==  22⋅3⋅59 2^{2} \cdot 3 \cdot 59
I4 I_4  == 1633516335 ==  33⋅5⋅112 3^{3} \cdot 5 \cdot 11^{2}
I6 I_6  == 35820903582090 ==  2⋅33⋅5⋅13267 2 \cdot 3^{3} \cdot 5 \cdot 13267
I10 I_{10}  == −16200-16200 ==  −23⋅34⋅52 - 2^{3} \cdot 3^{4} \cdot 5^{2}
J2 J_2  == 708708 ==  22⋅3⋅59 2^{2} \cdot 3 \cdot 59
J4 J_4  == 99969996 ==  22⋅3⋅72⋅17 2^{2} \cdot 3 \cdot 7^{2} \cdot 17
J6 J_6  == −220864-220864 ==  −26⋅7⋅17⋅29 - 2^{6} \cdot 7 \cdot 17 \cdot 29
J8 J_8  == −64072932-64072932 ==  −22⋅3⋅7⋅11⋅17⋅4079 - 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \cdot 4079
J10 J_{10}  == −129600-129600 ==  −26⋅34⋅52 - 2^{6} \cdot 3^{4} \cdot 5^{2}
g1 g_1  == −34316366352/25-34316366352/25
g2 g_2  == −684322828/25-684322828/25
g3 g_3  == 192206896/225192206896/225

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

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

Automorphism group

Aut(X)\mathrm{Aut}(X)≃\simeq C2C_2
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ‾)\mathrm{Aut}(X_{\overline{\Q}})≃\simeq C2C_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: 00

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

This curve is locally solvable except over Q3\Q_{3}.

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 9.1.34012224000000.6

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:00
Regulator: 1 1
Real period: 2.045372 2.045372
Tamagawa product: 1 1
Torsion order:1 1
Leading coefficient: 2.045372 2.045372
Analytic order of Ш: 1 1   (rounded)
Order of Ш:square

Local invariants

Prime ord(NN) ord(Δ\Delta) Tamagawa L-factor Cluster picture
22 66 66 11 1+T1 + T
33 44 44 11 1−3T+3T21 - 3 T + 3 T^{2}
55 22 22 11 1+2T+5T21 + 2 T + 5 T^{2}

Galois representations

The mod-ℓ\ell Galois representation has maximal image GSp(4,Fℓ)\GSp(4,\F_\ell) for all primes ℓ \ell except those listed.

Prime ℓ\ell mod-ℓ\ell image Is torsion prime?
22 2.40.1 no
33 3.40.1 no

Sato-Tate group

ST\mathrm{ST}≃\simeq USp(4)\mathrm{USp}(4)
ST0\mathrm{ST}^0≃\simeq USp(4)\mathrm{USp}(4)

Decomposition of the Jacobian

Simple over Q‾\overline{\Q}

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of GL2\GL_2-type over Q\Q

Endomorphism ring over Q\Q:

End⁡(J)\End (J_{})≃\simeqZ\Z
End⁡(J)⊗Q\End (J_{}) \otimes \Q ≃\simeqQ\Q
End⁡(J)⊗R\End (J_{}) \otimes \R≃\simeq R\R

All Q‾\overline{\Q}-endomorphisms of the Jacobian are defined over Q\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);