Properties

Label 588.a.18816.1
Conductor 588588
Discriminant 18816-18816
Mordell-Weil group Z/24Z\Z/{24}\Z
Sato-Tate group SU(2)×SU(2)\mathrm{SU}(2)\times\mathrm{SU}(2)
End(JQ)R\End(J_{\overline{\Q}}) \otimes \R R×R\R \times \R
End(JQ)Q\End(J_{\overline{\Q}}) \otimes \Q Q×Q\Q \times \Q
End(J)Q\End(J) \otimes \Q Q×Q\Q \times \Q
Q\overline{\Q}-simple no
GL2\mathrm{GL}_2-type yes

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Minimal equation

Minimal equation

Simplified equation

y2+(x3+1)y=x5+x4+5x2+12x+8y^2 + (x^3 + 1)y = x^5 + x^4 + 5x^2 + 12x + 8 (homogenize, simplify)
y2+(x3+z3)y=x5z+x4z2+5x2z4+12xz5+8z6y^2 + (x^3 + z^3)y = x^5z + x^4z^2 + 5x^2z^4 + 12xz^5 + 8z^6 (dehomogenize, simplify)
y2=x6+4x5+4x4+2x3+20x2+48x+33y^2 = x^6 + 4x^5 + 4x^4 + 2x^3 + 20x^2 + 48x + 33 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 12, 5, 0, 1, 1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![8, 12, 5, 0, 1, 1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([33, 48, 20, 2, 4, 4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  588588 == 22372 2^{2} \cdot 3 \cdot 7^{2}
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  18816-18816 == 27372 - 2^{7} \cdot 3 \cdot 7^{2}
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 748748 ==  221117 2^{2} \cdot 11 \cdot 17
I4 I_4  == 1154511545 ==  52309 5 \cdot 2309
I6 I_6  == 29027872902787 ==  2902787 2902787
I10 I_{10}  == 24084482408448 ==  214372 2^{14} \cdot 3 \cdot 7^{2}
J2 J_2  == 187187 ==  1117 11 \cdot 17
J4 J_4  == 976976 ==  2461 2^{4} \cdot 61
J6 J_6  == 192-192 ==  263 - 2^{6} \cdot 3
J8 J_8  == 247120-247120 ==  2453089 - 2^{4} \cdot 5 \cdot 3089
J10 J_{10}  == 1881618816 ==  27372 2^{7} \cdot 3 \cdot 7^{2}
g1 g_1  == 228669389707/18816228669389707/18816
g2 g_2  == 398891383/1176398891383/1176
g3 g_3  == 34969/98-34969/98

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 C22C_2^2
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ)\mathrm{Aut}(X_{\overline{\Q}})\simeq C22C_2^2
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: (1:0:0),(1:1:0),(1:1:1),(1:1:1),(2:3:1),(2:4:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-2 : 3 : 1),\, (-2 : 4 : 1)
All points: (1:0:0),(1:1:0),(1:1:1),(1:1:1),(2:3:1),(2:4:1)(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-2 : 3 : 1),\, (-2 : 4 : 1)
All points: (1:1:0),(1:1:0),(2:1:1),(2:1:1),(1:2:1),(1:2:1)(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : -1 : 1),\, (-2 : 1 : 1),\, (-1 : -2 : 1),\, (-1 : 2 : 1)

Copy content magma:[C![-2,3,1],C![-2,4,1],C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-2,-1,1],C![-2,1,1],C![-1,-2,1],C![-1,2,1],C![1,-1,0],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/24Z\Z/{24}\Z

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

Generator D0D_0 Height Order
(2:4:1)+(1:1:1)(1:1:0)(1:0:0)(-2 : 4 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x+z)(x+2z)(x + z) (x + 2z) == 0,0, yy == 5xz26z3-5xz^2 - 6z^3 00 2424
Generator D0D_0 Height Order
(2:4:1)+(1:1:1)(1:1:0)(1:0:0)(-2 : 4 : 1) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0) (x+z)(x+2z)(x + z) (x + 2z) == 0,0, yy == 5xz26z3-5xz^2 - 6z^3 00 2424
Generator D0D_0 Height Order
(2:1:1)+(1:2:1)(1:1:0)(1:1:0)(-2 : 1 : 1) + (-1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0) (x+z)(x+2z)(x + z) (x + 2z) == 0,0, yy == x310xz211z3x^3 - 10xz^2 - 11z^3 00 2424

2-torsion field: 8.0.12446784.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: 00
Mordell-Weil rank: 00
2-Selmer rank:11
Regulator: 1 1
Real period: 20.65814 20.65814
Tamagawa product: 8 8
Torsion order:24 24
Leading coefficient: 0.286918 0.286918
Analytic order of Ш: 1 1   (rounded)
Order of Ш:square

Local invariants

Prime ord(NN) ord(Δ\Delta) Tamagawa L-factor Cluster picture
22 22 77 88 (1T)(1+T)( 1 - T )( 1 + T )
33 11 11 11 (1+T)(1+2T+3T2)( 1 + T )( 1 + 2 T + 3 T^{2} )
77 22 22 11 (1T)(1+T)( 1 - T )( 1 + T )

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 2.45.1 yes
33 3.720.4 yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over Q\Q

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 42.a

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_{})\simeqan order of index 22 in Z×Z\Z \times \Z
End(J)Q\End (J_{}) \otimes \Q \simeqQ\Q ×\times Q\Q
End(J)R\End (J_{}) \otimes \R\simeq R×R\R \times \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);