Properties

Label 975.a.63375.1
Conductor 975975
Discriminant 63375-63375
Mordell-Weil group Z/2ZZ/6Z\Z/{2}\Z \oplus \Z/{6}\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+(x3+1)y=x5+x3+2x2+x1y^2 + (x^3 + 1)y = -x^5 + x^3 + 2x^2 + x - 1 (homogenize, simplify)
y2+(x3+z3)y=x5z+x3z3+2x2z4+xz5z6y^2 + (x^3 + z^3)y = -x^5z + x^3z^3 + 2x^2z^4 + xz^5 - z^6 (dehomogenize, simplify)
y2=x64x5+6x3+8x2+4x3y^2 = x^6 - 4x^5 + 6x^3 + 8x^2 + 4x - 3 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 2, 1, 0, -1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 2, 1, 0, -1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-3, 4, 8, 6, 0, -4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  975975 == 35213 3 \cdot 5^{2} \cdot 13
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  63375-63375 == 353132 - 3 \cdot 5^{3} \cdot 13^{2}
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 148148 ==  2237 2^{2} \cdot 37
I4 I_4  == 48575-48575 ==  522967 - 5^{2} \cdot 29 \cdot 67
I6 I_6  == 4076175-4076175 ==  3521723139 - 3 \cdot 5^{2} \cdot 17 \cdot 23 \cdot 139
I10 I_{10}  == 8112000-8112000 ==  27353132 - 2^{7} \cdot 3 \cdot 5^{3} \cdot 13^{2}
J2 J_2  == 3737 ==  37 37
J4 J_4  == 20812081 ==  2081 2081
J6 J_6  == 3592935929 ==  193161 19 \cdot 31 \cdot 61
J8 J_8  == 750297-750297 ==  3383653 - 3 \cdot 383 \cdot 653
J10 J_{10}  == 63375-63375 ==  353132 - 3 \cdot 5^{3} \cdot 13^{2}
g1 g_1  == 69343957/63375-69343957/63375
g2 g_2  == 105408893/63375-105408893/63375
g3 g_3  == 49186801/63375-49186801/63375

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),(1:1:0),(1:0:1),(3:14:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (3 : -14 : 1)
All points: (1:0:0),(1:1:0),(1:0:1),(3:14:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (3 : -14 : 1)
All points: (1:1:0),(1:1:0),(1:0:1),(3:0:1)(1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (3 : 0 : 1)

Copy content magma:[C![-1,0,1],C![1,-1,0],C![1,0,0],C![3,-14,1]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![1,-1,0],C![1,1,0],C![3,0,1]]; // simplified model
 

Number of rational Weierstrass points: 22

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/2ZZ/6Z\Z/{2}\Z \oplus \Z/{6}\Z

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

Generator D0D_0 Height Order
D0(1:1:0)(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) x23xz+z2x^2 - 3xz + z^2 == 0,0, yy == 4xz2+z3-4xz^2 + z^3 00 22
D0(1:1:0)(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) 2x22xz+z22x^2 - 2xz + z^2 == 0,0, 2y2y == 4xz2z34xz^2 - z^3 00 66
Generator D0D_0 Height Order
D0(1:1:0)(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) x23xz+z2x^2 - 3xz + z^2 == 0,0, yy == 4xz2+z3-4xz^2 + z^3 00 22
D0(1:1:0)(1:0:0)D_0 - (1 : -1 : 0) - (1 : 0 : 0) 2x22xz+z22x^2 - 2xz + z^2 == 0,0, 2y2y == 4xz2z34xz^2 - z^3 00 66
Generator D0D_0 Height Order
D0(1:1:0)(1:1:0)D_0 - (1 : -1 : 0) - (1 : 1 : 0) x23xz+z2x^2 - 3xz + z^2 == 0,0, yy == x38xz2+3z3x^3 - 8xz^2 + 3z^3 00 22
D0(1:1:0)(1:1:0)D_0 - (1 : -1 : 0) - (1 : 1 : 0) 2x22xz+z22x^2 - 2xz + z^2 == 0,0, 2y2y == x3+8xz2z3x^3 + 8xz^2 - z^3 00 66

2-torsion field: Q(3,5)\Q(\sqrt{-3}, \sqrt{5})

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:22
Regulator: 1 1
Real period: 14.35629 14.35629
Tamagawa product: 4 4
Torsion order:12 12
Leading coefficient: 0.398785 0.398785
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 11 11 1-1 (1T)(1+2T+3T2)( 1 - T )( 1 + 2 T + 3 T^{2} ) yes
55 22 33 22 1-1 1+5T21 + 5 T^{2} yes
1313 11 22 22 11 (1+T)(1+4T+13T2)( 1 + T )( 1 + 4 T + 13 T^{2} ) yes

Galois representations

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

Prime \ell mod-\ell image Is torsion prime?
22 2.180.3 yes
33 3.80.1 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);