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Properties

Label 893.a.893.1
Conductor 893893
Discriminant 893893
Mordell-Weil group Z\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+1)y=−x4−x2y^2 + (x^3 + x + 1)y = -x^4 - x^2 (homogenize, simplify)
y2+(x3+xz2+z3)y=−x4z2−x2z4y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 - x^2z^4 (dehomogenize, simplify)
y2=x6−2x4+2x3−3x2+2x+1y^2 = x^6 - 2x^4 + 2x^3 - 3x^2 + 2x + 1 (homogenize, minimize)

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

Invariants

Conductor: N N  ==  893893 == 19⋅47 19 \cdot 47
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  893893 == 19⋅47 19 \cdot 47
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 156156 ==  22⋅3⋅13 2^{2} \cdot 3 \cdot 13
I4 I_4  == −519-519 ==  −3⋅173 - 3 \cdot 173
I6 I_6  == −11805-11805 ==  −3⋅5⋅787 - 3 \cdot 5 \cdot 787
I10 I_{10}  == −114304-114304 ==  −27⋅19⋅47 - 2^{7} \cdot 19 \cdot 47
J2 J_2  == 3939 ==  3⋅13 3 \cdot 13
J4 J_4  == 8585 ==  5⋅17 5 \cdot 17
J6 J_6  == 6767 ==  67 67
J8 J_8  == −1153-1153 ==  −1153 -1153
J10 J_{10}  == −893-893 ==  −19⋅47 - 19 \cdot 47
g1 g_1  == −90224199/893-90224199/893
g2 g_2  == −5042115/893-5042115/893
g3 g_3  == −101907/893-101907/893

  (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) (1:−1:0)(1 : -1 : 0) (0:0:1)(0 : 0 : 1) (0:−1:1)(0 : -1 : 1) (1:−1:1)(1 : -1 : 1) (1:−2:1)(1 : -2 : 1)
(−2:4:1)(-2 : 4 : 1) (−2:5:1)(-2 : 5 : 1)
All points
(1:0:0)(1 : 0 : 0) (1:−1:0)(1 : -1 : 0) (0:0:1)(0 : 0 : 1) (0:−1:1)(0 : -1 : 1) (1:−1:1)(1 : -1 : 1) (1:−2:1)(1 : -2 : 1)
(−2:4:1)(-2 : 4 : 1) (−2:5:1)(-2 : 5 : 1)
All points
(1:−1:0)(1 : -1 : 0) (1:1:0)(1 : 1 : 0) (0:−1:1)(0 : -1 : 1) (0:1:1)(0 : 1 : 1) (1:−1:1)(1 : -1 : 1) (1:1:1)(1 : 1 : 1)
(−2:−1:1)(-2 : -1 : 1) (−2:1:1)(-2 : 1 : 1)

Copy content magma:[C![-2,4,1],C![-2,5,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-2,-1,1],C![-2,1,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: 00

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\Z

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

Generator D0D_0 Height Order
D0−2⋅(1:−1:0)D_0 - 2 \cdot(1 : -1 : 0) z2z^2 == 0,0, yy == 00 0.0064290.006429 ∞\infty
Generator D0D_0 Height Order
D0−2⋅(1:−1:0)D_0 - 2 \cdot(1 : -1 : 0) z2z^2 == 0,0, yy == 00 0.0064290.006429 ∞\infty
Generator D0D_0 Height Order
D0−2⋅(1:−1:0)D_0 - 2 \cdot(1 : -1 : 0) z2z^2 == 0,0, yy == x3+xz2+z3x^3 + xz^2 + z^3 0.0064290.006429 ∞\infty

2-torsion field: 6.2.57152.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: 11
Mordell-Weil rank: 11
2-Selmer rank:11
Regulator: 0.006429 0.006429
Real period: 23.40243 23.40243
Tamagawa product: 1 1
Torsion order:1 1
Leading coefficient: 0.150458 0.150458
Analytic order of Ш: 1 1   (rounded)
Order of Ш:square

Local invariants

Prime ord(NN) ord(Δ\Delta) Tamagawa L-factor Cluster picture
1919 11 11 11 (1−T)(1+5T+19T2)( 1 - T )( 1 + 5 T + 19 T^{2} )
4747 11 11 11 (1+T)(1+3T+47T2)( 1 + T )( 1 + 3 T + 47 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 .

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);