sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 0, -1, 1]), R([]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 0, -1, 1], R![]);
sage: X = HyperellipticCurve(R([0, -1, -1, 0, -1, 1]))
magma: X,pi:= SimplifiedModel(C);
Conductor : N N N = = = 25600 25600 2 5 6 0 0 = = = 2 10 ⋅ 5 2 2^{10} \cdot 5^{2} 2 1 0 ⋅ 5 2
magma: Conductor(LSeries(C)); Factorization($1);
Discriminant : Δ \Delta Δ = = = − 128000 -128000 − 1 2 8 0 0 0 = = = − 2 10 ⋅ 5 3 - 2^{10} \cdot 5^{3} − 2 1 0 ⋅ 5 3
magma: Discriminant(C); Factorization(Integers()!$1);
I 2 I_2 I 2 = = = 56 56 5 6 = = =
2 3 ⋅ 7 2^{3} \cdot 7 2 3 ⋅ 7
I 4 I_4 I 4 = = = − 80 -80 − 8 0 = = =
− 2 4 ⋅ 5 - 2^{4} \cdot 5 − 2 4 ⋅ 5
I 6 I_6 I 6 = = = − 260 -260 − 2 6 0 = = =
− 2 2 ⋅ 5 ⋅ 13 - 2^{2} \cdot 5 \cdot 13 − 2 2 ⋅ 5 ⋅ 1 3
I 10 I_{10} I 1 0 = = = 500 500 5 0 0 = = =
2 2 ⋅ 5 3 2^{2} \cdot 5^{3} 2 2 ⋅ 5 3
J 2 J_2 J 2 = = = 112 112 1 1 2 = = =
2 4 ⋅ 7 2^{4} \cdot 7 2 4 ⋅ 7
J 4 J_4 J 4 = = = 736 736 7 3 6 = = =
2 5 ⋅ 23 2^{5} \cdot 23 2 5 ⋅ 2 3
J 6 J_6 J 6 = = = − 1536 -1536 − 1 5 3 6 = = =
− 2 9 ⋅ 3 - 2^{9} \cdot 3 − 2 9 ⋅ 3
J 8 J_8 J 8 = = = − 178432 -178432 − 1 7 8 4 3 2 = = =
− 2 8 ⋅ 17 ⋅ 41 - 2^{8} \cdot 17 \cdot 41 − 2 8 ⋅ 1 7 ⋅ 4 1
J 10 J_{10} J 1 0 = = = 128000 128000 1 2 8 0 0 0 = = =
2 10 ⋅ 5 3 2^{10} \cdot 5^{3} 2 1 0 ⋅ 5 3
g 1 g_1 g 1 = = = 17210368 / 125 17210368/125 1 7 2 1 0 3 6 8 / 1 2 5
g 2 g_2 g 2 = = = 1009792 / 125 1009792/125 1 0 0 9 7 9 2 / 1 2 5
g 3 g_3 g 3 = = = − 18816 / 125 -18816/125 − 1 8 8 1 6 / 1 2 5
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
A u t ( X ) \mathrm{Aut}(X) A u t ( X ) ≃ \simeq ≃
C 4 C_4 C 4
magma: AutomorphismGroup(C); IdentifyGroup($1);
A u t ( X Q ‾ ) \mathrm{Aut}(X_{\overline{\Q}}) A u t ( X Q ) ≃ \simeq ≃
D 4 D_4 D 4
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
All points :
( 1 : 0 : 0 ) , ( 0 : 0 : 1 ) (1 : 0 : 0),\, (0 : 0 : 1) ( 1 : 0 : 0 ) , ( 0 : 0 : 1 )
All points :
( 1 : 0 : 0 ) , ( 0 : 0 : 1 ) (1 : 0 : 0),\, (0 : 0 : 1) ( 1 : 0 : 0 ) , ( 0 : 0 : 1 )
All points :
( 1 : 0 : 0 ) , ( 0 : 0 : 1 ) (1 : 0 : 0),\, (0 : 0 : 1) ( 1 : 0 : 0 ) , ( 0 : 0 : 1 )
magma: [C![0,0,1],C![1,0,0]]; // minimal model
magma: [C![0,0,1],C![1,0,0]]; // simplified model
Number of rational Weierstrass points : 2 2 2
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
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);
Group structure : Z / 2 Z ⊕ Z / 2 Z \Z/{2}\Z \oplus \Z/{2}\Z Z / 2 Z ⊕ Z / 2 Z
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator
D 0 D_0 D 0
Height
Order
D 0 − 2 ⋅ ( 1 : 0 : 0 ) D_0 - 2 \cdot(1 : 0 : 0) D 0 − 2 ⋅ ( 1 : 0 : 0 )
x 2 − x z − z 2 x^2 - xz - z^2 x 2 − x z − z 2
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
( 0 : 0 : 1 ) − ( 1 : 0 : 0 ) (0 : 0 : 1) - (1 : 0 : 0) ( 0 : 0 : 1 ) − ( 1 : 0 : 0 )
x x x
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
Generator
D 0 D_0 D 0
Height
Order
D 0 − 2 ⋅ ( 1 : 0 : 0 ) D_0 - 2 \cdot(1 : 0 : 0) D 0 − 2 ⋅ ( 1 : 0 : 0 )
x 2 − x z − z 2 x^2 - xz - z^2 x 2 − x z − z 2
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
( 0 : 0 : 1 ) − ( 1 : 0 : 0 ) (0 : 0 : 1) - (1 : 0 : 0) ( 0 : 0 : 1 ) − ( 1 : 0 : 0 )
x x x
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
Generator
D 0 D_0 D 0
Height
Order
D 0 − 2 ⋅ ( 1 : 0 : 0 ) D_0 - 2 \cdot(1 : 0 : 0) D 0 − 2 ⋅ ( 1 : 0 : 0 )
x 2 − x z − z 2 x^2 - xz - z^2 x 2 − x z − z 2
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
( 0 : 0 : 1 ) − ( 1 : 0 : 0 ) (0 : 0 : 1) - (1 : 0 : 0) ( 0 : 0 : 1 ) − ( 1 : 0 : 0 )
x x x
= = =
0 , 0, 0 ,
y y y
= = =
0 0 0
0 0 0
2 2 2
2-torsion field : Q ( i , 5 ) \Q(i, \sqrt{5}) Q ( i , 5 )
For primes ℓ ≥ 5 \ell \ge 5 ℓ ≥ 5 the Galois representation data has not been computed for this curve since it is not generic.
For primes ℓ ≤ 3 \ell \le 3 ℓ ≤ 3 , the image of the mod-ℓ \ell ℓ Galois representation is listed in the table below, whenever it is not all of GSp ( 4 , F ℓ ) \GSp(4,\F_\ell) GSp ( 4 , F ℓ ) .
S T \mathrm{ST} S T ≃ \simeq ≃ E 4 E_4 E 4
S T 0 \mathrm{ST}^0 S T 0 ≃ \simeq ≃ S U ( 2 ) \mathrm{SU}(2) S U ( 2 )
Splits over the number field Q ( b ) ≃ \Q (b) \simeq Q ( b ) ≃ Q ( ζ 20 ) + \Q(\zeta_{20})^+ Q ( ζ 2 0 ) + with defining polynomial: x 4 − 5 x 2 + 5 x^{4} - 5 x^{2} + 5 x 4 − 5 x 2 + 5
Decomposes up to isogeny as the square of the elliptic curve isogeny class: y 2 = x 3 − g 4 / 48 x − g 6 / 864 y^2 = x^3 - g_4 / 48 x - g_6 / 864 y 2 = x 3 − g 4 / 4 8 x − g 6 / 8 6 4 with g 4 = − 32303200 130321 b 3 + 61624800 130321 b 2 + 44011200 130321 b − 84012560 130321 g_4 = -\frac{32303200}{130321} b^{3} + \frac{61624800}{130321} b^{2} + \frac{44011200}{130321} b - \frac{84012560}{130321} g 4 = − 1 3 0 3 2 1 3 2 3 0 3 2 0 0 b 3 + 1 3 0 3 2 1 6 1 6 2 4 8 0 0 b 2 + 1 3 0 3 2 1 4 4 0 1 1 2 0 0 b − 1 3 0 3 2 1 8 4 0 1 2 5 6 0 g 6 = − 530918514880 47045881 b 3 + 1009835128000 47045881 b 2 + 734310067200 47045881 b − 1396419152000 47045881 g_6 = -\frac{530918514880}{47045881} b^{3} + \frac{1009835128000}{47045881} b^{2} + \frac{734310067200}{47045881} b - \frac{1396419152000}{47045881} g 6 = − 4 7 0 4 5 8 8 1 5 3 0 9 1 8 5 1 4 8 8 0 b 3 + 4 7 0 4 5 8 8 1 1 0 0 9 8 3 5 1 2 8 0 0 0 b 2 + 4 7 0 4 5 8 8 1 7 3 4 3 1 0 0 6 7 2 0 0 b − 4 7 0 4 5 8 8 1 1 3 9 6 4 1 9 1 5 2 0 0 0 Conductor norm: 4096
magma: HeuristicDecompositionFactors(C);
Of GL 2 \GL_2 GL 2 -type over Q \Q Q
Endomorphism ring over Q \Q Q :
End ( J ) \End (J_{}) E n d ( J ) ≃ \simeq ≃ Z [ − 1 ] \Z [\sqrt{-1}] Z [ − 1 ] End ( J ) ⊗ Q \End (J_{}) \otimes \Q E n d ( J ) ⊗ Q ≃ \simeq ≃ Q ( − 1 ) \Q(\sqrt{-1}) Q ( − 1 ) End ( J ) ⊗ R \End (J_{}) \otimes \R E n d ( J ) ⊗ R ≃ \simeq ≃ C \C C
Smallest field over which all endomorphisms are defined:
Galois number field K = Q ( a ) ≃ K = \Q (a) \simeq K = Q ( a ) ≃ Q ( ζ 20 ) + \Q(\zeta_{20})^+ Q ( ζ 2 0 ) + with defining polynomial x 4 − 5 x 2 + 5 x^{4} - 5 x^{2} + 5 x 4 − 5 x 2 + 5
Not of GL 2 \GL_2 GL 2 -type over Q ‾ \overline{\Q} Q
Endomorphism ring over Q ‾ \overline{\Q} Q :
End ( J Q ‾ ) \End (J_{\overline{\Q}}) E n d ( J Q ) ≃ \simeq ≃ a non-Eichler order of index 4 4 4 in a maximal order of End ( J Q ‾ ) ⊗ Q \End (J_{\overline{\Q}}) \otimes \Q E n d ( J Q ) ⊗ Q End ( J Q ‾ ) ⊗ Q \End (J_{\overline{\Q}}) \otimes \Q E n d ( J Q ) ⊗ Q ≃ \simeq ≃ M 2 ( \mathrm{M}_2( M 2 ( Q \Q Q ) ) ) End ( J Q ‾ ) ⊗ R \End (J_{\overline{\Q}}) \otimes \R E n d ( J Q ) ⊗ R ≃ \simeq ≃ M 2 ( R ) \mathrm{M}_2 (\R) M 2 ( R )
Over subfield F ≃ F \simeq F ≃ Q ( 5 ) \Q(\sqrt{5}) Q ( 5 ) with generator a 2 − 2 a^{2} - 2 a 2 − 2 with minimal polynomial x 2 − x − 1 x^{2} - x - 1 x 2 − x − 1 :
End ( J F ) \End (J_{F}) E n d ( J F ) ≃ \simeq ≃ Z [ − 1 ] \Z [\sqrt{-1}] Z [ − 1 ] End ( J F ) ⊗ Q \End (J_{F}) \otimes \Q E n d ( J F ) ⊗ Q ≃ \simeq ≃ Q ( − 1 ) \Q(\sqrt{-1}) Q ( − 1 ) End ( J F ) ⊗ R \End (J_{F}) \otimes \R E n d ( J F ) ⊗ R ≃ \simeq ≃ C \C C
Sato Tate group:
E 2 E_2 E 2 Of
GL 2 \GL_2 GL 2 -type, simple
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);