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Properties

Label 10023.a.30069.1
Conductor 1002310023
Discriminant 3006930069
Mordell-Weil group Z\Z
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+y=x5+2x4+x3−2x2y^2 + y = x^5 + 2x^4 + x^3 - 2x^2 (homogenize, simplify)
y2+z3y=x5z+2x4z2+x3z3−2x2z4y^2 + z^3y = x^5z + 2x^4z^2 + x^3z^3 - 2x^2z^4 (dehomogenize, simplify)
y2=4x5+8x4+4x3−8x2+1y^2 = 4x^5 + 8x^4 + 4x^3 - 8x^2 + 1 (homogenize, minimize)

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

Invariants

Conductor: N N  ==  1002310023 == 3⋅13⋅257 3 \cdot 13 \cdot 257
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  3006930069 == 32⋅13⋅257 3^{2} \cdot 13 \cdot 257
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 280280 ==  23⋅5⋅7 2^{3} \cdot 5 \cdot 7
I4 I_4  == −2048-2048 ==  −211 - 2^{11}
I6 I_6  == −91928-91928 ==  −23⋅11491 - 2^{3} \cdot 11491
I10 I_{10}  == 120276120276 ==  22⋅32⋅13⋅257 2^{2} \cdot 3^{2} \cdot 13 \cdot 257
J2 J_2  == 140140 ==  22⋅5⋅7 2^{2} \cdot 5 \cdot 7
J4 J_4  == 11581158 ==  2⋅3⋅193 2 \cdot 3 \cdot 193
J6 J_6  == 32923292 ==  22⋅823 2^{2} \cdot 823
J8 J_8  == −220021-220021 ==  −220021 -220021
J10 J_{10}  == 3006930069 ==  32⋅13⋅257 3^{2} \cdot 13 \cdot 257
g1 g_1  == 53782400000/3006953782400000/30069
g2 g_2  == 1059184000/100231059184000/10023
g3 g_3  == 64523200/3006964523200/30069

  (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 C2C_2
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ‾)\mathrm{Aut}(X_{\overline{\Q}})≃\simeq C2C_2
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: (1:0:0), (0:0:1), (0:−1:1), (1:1:1), (1:−2:1)(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)
All points: (1:0:0), (0:0:1), (0:−1:1), (1:1:1), (1:−2:1)(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 1)
All points: (1:0:0), (0:−1:1), (0:1:1), (1:−3:1), (1:3:1)(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -3 : 1),\, (1 : 3 : 1)

Copy content magma:[C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1]]; // minimal model
 
Copy content magma:[C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,0,0],C![1,3,1]]; // simplified model
 

Number of rational Weierstrass points: 11

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\Z

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

Generator D0D_0 Height Order
(0:−1:1)+(1:1:1)−2⋅(1:0:0)(0 : -1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0) x(x−z)x (x - z) == 0,0, yy == 2xz2−z32xz^2 - z^3 0.0273830.027383 ∞\infty
Generator D0D_0 Height Order
(0:−1:1)+(1:1:1)−2⋅(1:0:0)(0 : -1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0) x(x−z)x (x - z) == 0,0, yy == 2xz2−z32xz^2 - z^3 0.0273830.027383 ∞\infty
Generator D0D_0 Height Order
(0:−1:1)+(1:3:1)−2⋅(1:0:0)(0 : -1 : 1) + (1 : 3 : 1) - 2 \cdot(1 : 0 : 0) x(x−z)x (x - z) == 0,0, yy == 4xz2−z34xz^2 - z^3 0.0273830.027383 ∞\infty

2-torsion field: 5.1.53456.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 11
Mordell-Weil rank: 11
2-Selmer rank:11
Regulator: 0.027383 0.027383
Real period: 11.96499 11.96499
Tamagawa product: 2 2
Torsion order:1 1
Leading coefficient: 0.655294 0.655294
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?
33 11 22 22 11 (1+T)(1−T+3T2)( 1 + T )( 1 - T + 3 T^{2} ) yes
1313 11 11 11 11 (1+T)(1−T+13T2)( 1 + T )( 1 - T + 13 T^{2} ) yes
257257 11 11 11 −1-1 (1−T)(1+24T+257T2)( 1 - T )( 1 + 24 T + 257 T^{2} ) yes

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 2.6.1 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}

Copy content 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.

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);