Properties

Label 25600.e.128000.1
Conductor 2560025600
Discriminant 128000-128000
Mordell-Weil group Z/2ZZ/2Z\Z/{2}\Z \oplus \Z/{2}\Z
Sato-Tate group E4E_4
End(JQ)R\End(J_{\overline{\Q}}) \otimes \R M2(R)\mathrm{M}_2(\R)
End(JQ)Q\End(J_{\overline{\Q}}) \otimes \Q M2(Q)\mathrm{M}_2(\Q)
End(J)Q\End(J) \otimes \Q CM\mathsf{CM}
Q\overline{\Q}-simple no
GL2\mathrm{GL}_2-type yes

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

y2=x5x4x2xy^2 = x^5 - x^4 - x^2 - x (homogenize, simplify)
y2=x5zx4z2x2z4xz5y^2 = x^5z - x^4z^2 - x^2z^4 - xz^5 (dehomogenize, simplify)
y2=x5x4x2xy^2 = x^5 - x^4 - x^2 - x (homogenize, minimize)

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

Invariants

Conductor: N N  ==  2560025600 == 21052 2^{10} \cdot 5^{2}
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  128000-128000 == 21053 - 2^{10} \cdot 5^{3}
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 5656 ==  237 2^{3} \cdot 7
I4 I_4  == 80-80 ==  245 - 2^{4} \cdot 5
I6 I_6  == 260-260 ==  22513 - 2^{2} \cdot 5 \cdot 13
I10 I_{10}  == 500500 ==  2253 2^{2} \cdot 5^{3}
J2 J_2  == 112112 ==  247 2^{4} \cdot 7
J4 J_4  == 736736 ==  2523 2^{5} \cdot 23
J6 J_6  == 1536-1536 ==  293 - 2^{9} \cdot 3
J8 J_8  == 178432-178432 ==  281741 - 2^{8} \cdot 17 \cdot 41
J10 J_{10}  == 128000128000 ==  21053 2^{10} \cdot 5^{3}
g1 g_1  == 17210368/12517210368/125
g2 g_2  == 1009792/1251009792/125
g3 g_3  == 18816/125-18816/125

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 C4C_4
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ)\mathrm{Aut}(X_{\overline{\Q}})\simeq D4D_4
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

Copy content magma:[C![0,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![0,0,1],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/2ZZ/2Z\Z/{2}\Z \oplus \Z/{2}\Z

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

Generator D0D_0 Height Order
D02(1:0:0)D_0 - 2 \cdot(1 : 0 : 0) x2xzz2x^2 - xz - z^2 == 0,0, yy == 00 00 22
(0:0:1)(1:0:0)(0 : 0 : 1) - (1 : 0 : 0) xx == 0,0, yy == 00 00 22
Generator D0D_0 Height Order
D02(1:0:0)D_0 - 2 \cdot(1 : 0 : 0) x2xzz2x^2 - xz - z^2 == 0,0, yy == 00 00 22
(0:0:1)(1:0:0)(0 : 0 : 1) - (1 : 0 : 0) xx == 0,0, yy == 00 00 22
Generator D0D_0 Height Order
D02(1:0:0)D_0 - 2 \cdot(1 : 0 : 0) x2xzz2x^2 - xz - z^2 == 0,0, yy == 00 00 22
(0:0:1)(1:0:0)(0 : 0 : 1) - (1 : 0 : 0) xx == 0,0, yy == 00 00 22

2-torsion field: Q(i,5)\Q(i, \sqrt{5})

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:22
Regulator: 1 1
Real period: 9.947770 9.947770
Tamagawa product: 2 2
Torsion order:4 4
Leading coefficient: 1.243471 1.243471
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?
22 1010 1010 11 1-1^* 11 no
55 22 33 22 1-1 12T+5T21 - 2 T + 5 T^{2} yes

Galois representations

For primes 5\ell \ge 5 the Galois representation data has not been computed for this curve since it is not generic.

For primes 3\ell \le 3, the image of the mod-\ell Galois representation is listed in the table below, whenever it is not all of GSp(4,F)\GSp(4,\F_\ell).

Prime \ell mod-\ell image Is torsion prime?
22 2.180.3 yes
33 3.540.6 no

Sato-Tate group

ST\mathrm{ST}\simeq E4E_4
ST0\mathrm{ST}^0\simeq SU(2)\mathrm{SU}(2)

Decomposition of the Jacobian

Splits over the number field Q(b)\Q (b) \simeq Q(ζ20)+\Q(\zeta_{20})^+ with defining polynomial:
  x45x2+5x^{4} - 5 x^{2} + 5

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  y2=x3g4/48xg6/864y^2 = x^3 - g_4 / 48 x - g_6 / 864 with
  g4=32303200130321b3+61624800130321b2+44011200130321b84012560130321g_4 = -\frac{32303200}{130321} b^{3} + \frac{61624800}{130321} b^{2} + \frac{44011200}{130321} b - \frac{84012560}{130321}
  g6=53091851488047045881b3+100983512800047045881b2+73431006720047045881b139641915200047045881g_6 = -\frac{530918514880}{47045881} b^{3} + \frac{1009835128000}{47045881} b^{2} + \frac{734310067200}{47045881} b - \frac{1396419152000}{47045881}
   Conductor norm: 4096

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of GL2\GL_2-type over Q\Q

Endomorphism ring over Q\Q:

End(J)\End (J_{})\simeqZ[1]\Z [\sqrt{-1}]
End(J)Q\End (J_{}) \otimes \Q \simeqQ(1)\Q(\sqrt{-1})
End(J)R\End (J_{}) \otimes \R\simeq C\C

Smallest field over which all endomorphisms are defined:
Galois number field K=Q(a)K = \Q (a) \simeq Q(ζ20)+\Q(\zeta_{20})^+ with defining polynomial x45x2+5x^{4} - 5 x^{2} + 5

Not of GL2\GL_2-type over Q\overline{\Q}

Endomorphism ring over Q\overline{\Q}:

End(JQ)\End (J_{\overline{\Q}})\simeqa non-Eichler order of index 44 in a maximal order of End(JQ)Q\End (J_{\overline{\Q}}) \otimes \Q
End(JQ)Q\End (J_{\overline{\Q}}) \otimes \Q \simeqM2(\mathrm{M}_2(Q\Q))
End(JQ)R\End (J_{\overline{\Q}}) \otimes \R\simeq M2(R)\mathrm{M}_2 (\R)

Remainder of the endomorphism lattice by field

Over subfield FF \simeq Q(5)\Q(\sqrt{5}) with generator a22a^{2} - 2 with minimal polynomial x2x1x^{2} - x - 1:

End(JF)\End (J_{F})\simeqZ[1]\Z [\sqrt{-1}]
End(JF)Q\End (J_{F}) \otimes \Q \simeqQ(1)\Q(\sqrt{-1})
End(JF)R\End (J_{F}) \otimes \R\simeq C\C
  Sato Tate group: E2E_2
  Of GL2\GL_2-type, simple

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);