Properties

Label 1047.a.3141.1
Conductor 10471047
Discriminant 31413141
Mordell-Weil group Z/10Z\Z/{10}\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+x)y=xy^2 + (x^3 + x)y = x (homogenize, simplify)
y2+(x3+xz2)y=xz5y^2 + (x^3 + xz^2)y = xz^5 (dehomogenize, simplify)
y2=x6+2x4+x2+4xy^2 = x^6 + 2x^4 + x^2 + 4x (homogenize, minimize)

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

Invariants

Conductor: N N  ==  10471047 == 3349 3 \cdot 349
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  31413141 == 32349 3^{2} \cdot 349
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 88 ==  23 2^{3}
I4 I_4  == 604604 ==  22151 2^{2} \cdot 151
I6 I_6  == 10171017 ==  32113 3^{2} \cdot 113
I10 I_{10}  == 12564-12564 ==  2232349 - 2^{2} \cdot 3^{2} \cdot 349
J2 J_2  == 44 ==  22 2^{2}
J4 J_4  == 100-100 ==  2252 - 2^{2} \cdot 5^{2}
J6 J_6  == 1-1 ==  1 -1
J8 J_8  == 2501-2501 ==  4161 - 41 \cdot 61
J10 J_{10}  == 3141-3141 ==  32349 - 3^{2} \cdot 349
g1 g_1  == 1024/3141-1024/3141
g2 g_2  == 6400/31416400/3141
g3 g_3  == 16/314116/3141

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

Copy content magma:[C![-1,1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // 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/10Z\Z/{10}\Z

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

Generator D0D_0 Height Order
(1:1:1)(1:1:0)(-1 : 1 : 1) - (1 : -1 : 0) z(x+z)z (x + z) == 0,0, yy == z3z^3 00 1010
Generator D0D_0 Height Order
(1:1:1)(1:1:0)(-1 : 1 : 1) - (1 : -1 : 0) z(x+z)z (x + z) == 0,0, yy == z3z^3 00 1010
Generator D0D_0 Height Order
(1:0:1)(1:1:0)(-1 : 0 : 1) - (1 : -1 : 0) z(x+z)z (x + z) == 0,0, yy == x3+xz2+2z3x^3 + xz^2 + 2z^3 00 1010

2-torsion field: 4.0.1396.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:11
Regulator: 1 1
Real period: 17.82167 17.82167
Tamagawa product: 2 2
Torsion order:10 10
Leading coefficient: 0.356433 0.356433
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 1-1 (1T)(1+T+3T2)( 1 - T )( 1 + T + 3 T^{2} ) yes
349349 11 11 11 1-1 (1T)(1+10T+349T2)( 1 - T )( 1 + 10 T + 349 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
55 not computed 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);