Properties

Label 1051.b.1051.1
Conductor 10511051
Discriminant 1051-1051
Mordell-Weil group Z/8Z\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+(x+1)y=x5x4y^2 + (x + 1)y = -x^5 - x^4 (homogenize, simplify)
y2+(xz2+z3)y=x5zx4z2y^2 + (xz^2 + z^3)y = -x^5z - x^4z^2 (dehomogenize, simplify)
y2=4x54x4+x2+2x+1y^2 = -4x^5 - 4x^4 + x^2 + 2x + 1 (homogenize, minimize)

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

Invariants

Conductor: N N  ==  10511051 == 1051 1051
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  1051-1051 == 1051 -1051
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 6464 ==  26 2^{6}
I4 I_4  == 200-200 ==  2352 - 2^{3} \cdot 5^{2}
I6 I_6  == 185185 ==  537 5 \cdot 37
I10 I_{10}  == 42044204 ==  221051 2^{2} \cdot 1051
J2 J_2  == 3232 ==  25 2^{5}
J4 J_4  == 7676 ==  2219 2^{2} \cdot 19
J6 J_6  == 241-241 ==  241 -241
J8 J_8  == 3372-3372 ==  223281 - 2^{2} \cdot 3 \cdot 281
J10 J_{10}  == 10511051 ==  1051 1051
g1 g_1  == 33554432/105133554432/1051
g2 g_2  == 2490368/10512490368/1051
g3 g_3  == 246784/1051-246784/1051

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

Copy content magma:[C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0]]; // 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/8Z\Z/{8}\Z

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

Generator D0D_0 Height Order
(0:0:1)(1:0:0)(0 : 0 : 1) - (1 : 0 : 0) xx == 0,0, yy == 00 00 88
Generator D0D_0 Height Order
(0:0:1)(1:0:0)(0 : 0 : 1) - (1 : 0 : 0) xx == 0,0, yy == 00 00 88
Generator D0D_0 Height Order
(0:1:1)(1:0:0)(0 : 1 : 1) - (1 : 0 : 0) xx == 0,0, yy == xz2+z3xz^2 + z^3 00 88

2-torsion field: 4.2.4204.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:11
Regulator: 1 1
Real period: 23.33171 23.33171
Tamagawa product: 1 1
Torsion order:8 8
Leading coefficient: 0.364558 0.364558
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?
10511051 11 11 11 11 (1+T)(128T+1051T2)( 1 + T )( 1 - 28 T + 1051 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 2.30.3 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);