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Properties

Label 461.a.461.2
Conductor 461461
Discriminant 461461
Mordell-Weil group trivial
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−x4−39x3+10x2+272x−306y^2 + y = x^5 - x^4 - 39x^3 + 10x^2 + 272x - 306 (homogenize, simplify)
y2+z3y=x5z−x4z2−39x3z3+10x2z4+272xz5−306z6y^2 + z^3y = x^5z - x^4z^2 - 39x^3z^3 + 10x^2z^4 + 272xz^5 - 306z^6 (dehomogenize, simplify)
y2=4x5−4x4−156x3+40x2+1088x−1223y^2 = 4x^5 - 4x^4 - 156x^3 + 40x^2 + 1088x - 1223 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-306, 272, 10, -39, -1, 1]), R([1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-306, 272, 10, -39, -1, 1], R![1]);
 
Copy content sage:X = HyperellipticCurve(R([-1223, 1088, 40, -156, -4, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  461461 == 461 461
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  461461 == 461 461
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 8066480664 ==  23⋅3⋅3361 2^{3} \cdot 3 \cdot 3361
I4 I_4  == 166117104166117104 ==  24⋅32⋅29⋅39779 2^{4} \cdot 3^{2} \cdot 29 \cdot 39779
I6 I_6  == 37527259529523752725952952 ==  23⋅32⋅71⋅1319⋅556559 2^{3} \cdot 3^{2} \cdot 71 \cdot 1319 \cdot 556559
I10 I_{10}  == 18441844 ==  22⋅461 2^{2} \cdot 461
J2 J_2  == 4033240332 ==  22⋅3⋅3361 2^{2} \cdot 3 \cdot 3361
J4 J_4  == 4009174240091742 ==  2⋅32⋅31⋅71849 2 \cdot 3^{2} \cdot 31 \cdot 71849
J6 J_6  == 4507573727645075737276 ==  22⋅83⋅135770293 2^{2} \cdot 83 \cdot 135770293
J8 J_8  == 5266171480526752661714805267 ==  3⋅613⋅8819⋅3247087 3 \cdot 613 \cdot 8819 \cdot 3247087
J10 J_{10}  == 461461 ==  461 461
g1 g_1  == 106720731303787612818432/461106720731303787612818432/461
g2 g_2  == 2630293443843585469056/4612630293443843585469056/461
g3 g_3  == 73323359651716069824/46173323359651716069824/461

  (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)(1 : 0 : 0)
All points: (1:0:0)(1 : 0 : 0)
All points: (1:0:0)(1 : 0 : 0)

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

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

2-torsion field: 5.1.7376.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:00
Regulator: 1 1
Real period: 0.245886 0.245886
Tamagawa product: 1 1
Torsion order:1 1
Leading coefficient: 0.245886 0.245886
Analytic order of Ш: 1 1   (rounded)
Order of Ш:square

Local invariants

Prime ord(NN) ord(Δ\Delta) Tamagawa L-factor Cluster picture
461461 11 11 11 (1+T)(1+461T2)( 1 + T )( 1 + 461 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 2.6.1 no
77 not computed 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);