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Properties

Label 472.a.944.1
Conductor 472472
Discriminant −944-944
Mordell-Weil group Z/2Z⊕Z/8Z\Z/{2}\Z \oplus \Z/{8}\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+(x2+1)y=x5−x4−2x3+xy^2 + (x^2 + 1)y = x^5 - x^4 - 2x^3 + x (homogenize, simplify)
y2+(x2z+z3)y=x5z−x4z2−2x3z3+xz5y^2 + (x^2z + z^3)y = x^5z - x^4z^2 - 2x^3z^3 + xz^5 (dehomogenize, simplify)
y2=4x5−3x4−8x3+2x2+4x+1y^2 = 4x^5 - 3x^4 - 8x^3 + 2x^2 + 4x + 1 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -2, -1, 1]), R([1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -2, -1, 1], R![1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 4, 2, -8, -3, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  472472 == 23⋅59 2^{3} \cdot 59
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  −944-944 == −24⋅59 - 2^{4} \cdot 59
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  == 760760 ==  23⋅5⋅19 2^{3} \cdot 5 \cdot 19
I6 I_6  == 6060460604 ==  22⋅109⋅139 2^{2} \cdot 109 \cdot 139
I10 I_{10}  == −3776-3776 ==  −26⋅59 - 2^{6} \cdot 59
J2 J_2  == 140140 ==  22⋅5⋅7 2^{2} \cdot 5 \cdot 7
J4 J_4  == 690690 ==  2⋅3⋅5⋅23 2 \cdot 3 \cdot 5 \cdot 23
J6 J_6  == 45444544 ==  26⋅71 2^{6} \cdot 71
J8 J_8  == 4001540015 ==  5⋅53⋅151 5 \cdot 53 \cdot 151
J10 J_{10}  == −944-944 ==  −24⋅59 - 2^{4} \cdot 59
g1 g_1  == −3361400000/59-3361400000/59
g2 g_2  == −118335000/59-118335000/59
g3 g_3  == −5566400/59-5566400/59

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

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

Number of rational Weierstrass points: 33

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

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

Generator D0D_0 Height Order
(−1:−1:1)−(1:0:0)(-1 : -1 : 1) - (1 : 0 : 0) x+zx + z == 0,0, yy == −z3-z^3 00 22
(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 == −z3-z^3 00 88
Generator D0D_0 Height Order
(−1:−1:1)−(1:0:0)(-1 : -1 : 1) - (1 : 0 : 0) x+zx + z == 0,0, yy == −z3-z^3 00 22
(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 == −z3-z^3 00 88
Generator D0D_0 Height Order
(−1:0:1)−(1:0:0)(-1 : 0 : 1) - (1 : 0 : 0) x+zx + z == 0,0, yy == x2z−z3x^2z - z^3 00 22
(0:−1:1)+(1:0:1)−2⋅(1:0:0)(0 : -1 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0) x(x−z)x (x - z) == 0,0, yy == x2z−z3x^2z - z^3 00 88

2-torsion field: 3.1.59.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:22
Regulator: 1 1
Real period: 29.11327 29.11327
Tamagawa product: 2 2
Torsion order:16 16
Leading coefficient: 0.227447 0.227447
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 33 44 22 1+T+2T21 + T + 2 T^{2}
5959 11 11 11 (1+T)(1−12T+59T2)( 1 + T )( 1 - 12 T + 59 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 2.120.3 yes

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);