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Properties

Label 360.a.6480.1
Conductor 360360
Discriminant 64806480
Mordell-Weil group Z/2Z⊕Z/2Z⊕Z/8Z\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{8}\Z
Sato-Tate group SU(2)×SU(2)\mathrm{SU}(2)\times\mathrm{SU}(2)
End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R R×R\R \times \R
End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q Q×Q\Q \times \Q
End⁡(J)⊗Q\End(J) \otimes \Q Q×Q\Q \times \Q
Q‾\overline{\Q}-simple no
GL2\mathrm{GL}_2-type yes

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Minimal equation

Minimal equation

Simplified equation

y2+(x3+x)y=−3x4+7x2−5y^2 + (x^3 + x)y = -3x^4 + 7x^2 - 5 (homogenize, simplify)
y2+(x3+xz2)y=−3x4z2+7x2z4−5z6y^2 + (x^3 + xz^2)y = -3x^4z^2 + 7x^2z^4 - 5z^6 (dehomogenize, simplify)
y2=x6−10x4+29x2−20y^2 = x^6 - 10x^4 + 29x^2 - 20 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 0, 7, 0, -3]), R([0, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 0, 7, 0, -3], R![0, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-20, 0, 29, 0, -10, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  360360 == 23⋅32⋅5 2^{3} \cdot 3^{2} \cdot 5
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  64806480 == 24⋅34⋅5 2^{4} \cdot 3^{4} \cdot 5
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 23602360 ==  23⋅5⋅59 2^{3} \cdot 5 \cdot 59
I4 I_4  == 1199211992 ==  23⋅1499 2^{3} \cdot 1499
I6 I_6  == 90478209047820 ==  22⋅3⋅5⋅150797 2^{2} \cdot 3 \cdot 5 \cdot 150797
I10 I_{10}  == 2592025920 ==  26⋅34⋅5 2^{6} \cdot 3^{4} \cdot 5
J2 J_2  == 11801180 ==  22⋅5⋅59 2^{2} \cdot 5 \cdot 59
J4 J_4  == 5601856018 ==  2⋅37⋅757 2 \cdot 37 \cdot 757
J6 J_6  == 34531203453120 ==  26⋅32⋅5⋅11⋅109 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 109
J8 J_8  == 234166319234166319 ==  3299⋅70981 3299 \cdot 70981
J10 J_{10}  == 64806480 ==  24⋅34⋅5 2^{4} \cdot 3^{4} \cdot 5
g1 g_1  == 28596971960000/8128596971960000/81
g2 g_2  == 1150492082200/811150492082200/81
g3 g_3  == 6677950400/96677950400/9

  (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

Aut(X)\mathrm{Aut}(X)≃\simeq C22C_2^2
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ‾)\mathrm{Aut}(X_{\overline{\Q}})≃\simeq C22C_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), (−2:5:1), (2:−5:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1),\, (-2 : 5 : 1),\, (2 : -5 : 1)
All points: (1:0:0), (1:−1:0), (−1:1:1), (1:−1:1), (−2:5:1), (2:−5:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1),\, (-2 : 5 : 1),\, (2 : -5 : 1)
All points: (1:−1:0), (1:1:0), (−1:0:1), (1:0:1), (−2:0:1), (2:0:1)(1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1),\, (-2 : 0 : 1),\, (2 : 0 : 1)

Copy content magma:[C![-2,5,1],C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-5,1]]; // minimal model
 
Copy content magma:[C![-2,0,1],C![-1,0,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,0,1]]; // simplified model
 

Number of rational Weierstrass points: 44

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/2Z⊕Z/2Z⊕Z/8Z\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{8}\Z

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

Generator D0D_0 Height Order
(1:−1:1)+(2:−5:1)−(1:−1:0)−(1:0:0)(1 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x−2z)(x−z)(x - 2z) (x - z) == 0,0, yy == −4xz2+3z3-4xz^2 + 3z^3 00 22
(−2:5:1)+(2:−5:1)−(1:−1:0)−(1:0:0)(-2 : 5 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x−2z)(x+2z)(x - 2z) (x + 2z) == 0,0, 2y2y == −5xz2-5xz^2 00 22
D0−(1:−1:0)−(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) x2−xz−3z2x^2 - xz - 3z^2 == 0,0, yy == −2xz2−2z3-2xz^2 - 2z^3 00 88
Generator D0D_0 Height Order
(1:−1:1)+(2:−5:1)−(1:−1:0)−(1:0:0)(1 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x−2z)(x−z)(x - 2z) (x - z) == 0,0, yy == −4xz2+3z3-4xz^2 + 3z^3 00 22
(−2:5:1)+(2:−5:1)−(1:−1:0)−(1:0:0)(-2 : 5 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x−2z)(x+2z)(x - 2z) (x + 2z) == 0,0, 2y2y == −5xz2-5xz^2 00 22
D0−(1:−1:0)−(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) x2−xz−3z2x^2 - xz - 3z^2 == 0,0, yy == −2xz2−2z3-2xz^2 - 2z^3 00 88
Generator D0D_0 Height Order
(1:0:1)+(2:0:1)−(1:−1:0)−(1:1:0)(1 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0) (x−2z)(x−z)(x - 2z) (x - z) == 0,0, yy == x3−7xz2+6z3x^3 - 7xz^2 + 6z^3 00 22
D0−(1:−1:0)−(1:1:0)D_0 - (1 : -1 : 0) - (1 : 1 : 0) (x−2z)(x+2z)(x - 2z) (x + 2z) == 0,0, 2y2y == x3−9xz2x^3 - 9xz^2 00 22
D0−(1:−1:0)−(1:1:0)D_0 - (1 : -1 : 0) - (1 : 1 : 0) x2−xz−3z2x^2 - xz - 3z^2 == 0,0, yy == x3−3xz2−4z3x^3 - 3xz^2 - 4z^3 00 88

2-torsion field: Q(5)\Q(\sqrt{5})

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:33
Regulator: 1 1
Real period: 24.16337 24.16337
Tamagawa product: 8 8
Torsion order:32 32
Leading coefficient: 0.188776 0.188776
Analytic order of Ш: 1 1   (rounded)
Order of Ш:square

Local invariants

Prime ord(NN) ord(Δ\Delta) Tamagawa Root number* L-factor Cluster picture Tame reduction?
22 33 44 22 −1∗-1^* 1+T+2T21 + T + 2 T^{2} no
33 22 44 44 11 (1+T)2( 1 + T )^{2} yes
55 11 11 11 −1-1 (1−T)(1+2T+5T2)( 1 - T )( 1 + 2 T + 5 T^{2} ) yes

Galois representations

For primes ℓ≥5\ell \ge 5 the Galois representation data has not been computed for this curve since it is not generic.

For primes ℓ≤3\ell \le 3, the image of the mod-ℓ\ell Galois representation is listed in the table below, whenever it is not all of GSp(4,Fℓ)\GSp(4,\F_\ell).

Prime ℓ\ell mod-ℓ\ell image Is torsion prime?
22 2.360.2 yes
33 3.90.1 no

Sato-Tate group

ST\mathrm{ST}≃\simeq SU(2)×SU(2)\mathrm{SU}(2)\times\mathrm{SU}(2)
ST0\mathrm{ST}^0≃\simeq SU(2)×SU(2)\mathrm{SU}(2)\times\mathrm{SU}(2)

Decomposition of the Jacobian

Splits over Q\Q

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 15.a
  Elliptic curve isogeny class 24.a

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of GL2\GL_2-type over Q\Q

Endomorphism ring over Q\Q:

End⁡(J)\End (J_{})≃\simeqan order of index 22 in Z×Z\Z \times \Z
End⁡(J)⊗Q\End (J_{}) \otimes \Q ≃\simeqQ\Q ×\times Q\Q
End⁡(J)⊗R\End (J_{}) \otimes \R≃\simeq R×R\R \times \R

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