Properties

Label 21316.a.42632.1
Conductor 2131621316
Discriminant 42632-42632
Mordell-Weil group ZZ\Z \oplus \Z
Sato-Tate group E6E_6
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+(x3+x+1)y=3x3+4x2+xy^2 + (x^3 + x + 1)y = 3x^3 + 4x^2 + x (homogenize, simplify)
y2+(x3+xz2+z3)y=3x3z3+4x2z4+xz5y^2 + (x^3 + xz^2 + z^3)y = 3x^3z^3 + 4x^2z^4 + xz^5 (dehomogenize, simplify)
y2=x6+2x4+14x3+17x2+6x+1y^2 = x^6 + 2x^4 + 14x^3 + 17x^2 + 6x + 1 (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 4, 3]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 4, 3], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 6, 17, 14, 2, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: N N  ==  2131621316 == 22732 2^{2} \cdot 73^{2}
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: Δ \Delta  ==  42632-42632 == 23732 - 2^{3} \cdot 73^{2}
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

I2 I_2  == 196196 ==  2272 2^{2} \cdot 7^{2}
I4 I_4  == 1233712337 ==  13273 13^{2} \cdot 73
I6 I_6  == 588745588745 ==  5731613 5 \cdot 73 \cdot 1613
I10 I_{10}  == 5456896-5456896 ==  210732 - 2^{10} \cdot 73^{2}
J2 J_2  == 4949 ==  72 7^{2}
J4 J_4  == 414-414 ==  23223 - 2 \cdot 3^{2} \cdot 23
J6 J_6  == 908-908 ==  22227 - 2^{2} \cdot 227
J8 J_8  == 53972-53972 ==  22103131 - 2^{2} \cdot 103 \cdot 131
J10 J_{10}  == 42632-42632 ==  23732 - 2^{3} \cdot 73^{2}
g1 g_1  == 282475249/42632-282475249/42632
g2 g_2  == 24353343/2131624353343/21316
g3 g_3  == 545027/10658545027/10658

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 C6C_6
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
Aut(XQ)\mathrm{Aut}(X_{\overline{\Q}})\simeq D6D_6
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
(1:0:0)(1 : 0 : 0) (1:1:0)(1 : -1 : 0) (0:0:1)(0 : 0 : 1) (1:0:1)(-1 : 0 : 1) (0:1:1)(0 : -1 : 1) (1:1:1)(-1 : 1 : 1)
(1:0:3)(-1 : 0 : 3) (2:3:1)(2 : 3 : 1) (3:7:2)(-3 : 7 : 2) (2:14:1)(2 : -14 : 1) (1:17:3)(-1 : -17 : 3) (3:24:2)(-3 : 24 : 2)
Known points
(1:0:0)(1 : 0 : 0) (1:1:0)(1 : -1 : 0) (0:0:1)(0 : 0 : 1) (1:0:1)(-1 : 0 : 1) (0:1:1)(0 : -1 : 1) (1:1:1)(-1 : 1 : 1)
(1:0:3)(-1 : 0 : 3) (2:3:1)(2 : 3 : 1) (3:7:2)(-3 : 7 : 2) (2:14:1)(2 : -14 : 1) (1:17:3)(-1 : -17 : 3) (3:24:2)(-3 : 24 : 2)
Known 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)
(2:17:1)(2 : -17 : 1) (2:17:1)(2 : 17 : 1) (1:17:3)(-1 : -17 : 3) (1:17:3)(-1 : 17 : 3) (3:17:2)(-3 : -17 : 2) (3:17:2)(-3 : 17 : 2)

Copy content magma:[C![-3,7,2],C![-3,24,2],C![-1,-17,3],C![-1,0,1],C![-1,0,3],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0],C![2,-14,1],C![2,3,1]]; // minimal model
 
Copy content magma:[C![-3,-17,2],C![-3,17,2],C![-1,-17,3],C![-1,-1,1],C![-1,17,3],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0],C![2,-17,1],C![2,17,1]]; // 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: ZZ\Z \oplus \Z

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

Generator D0D_0 Height Order
(0:1:1)(1:1:0)(0 : -1 : 1) - (1 : -1 : 0) zxz x == 0,0, yy == z3-z^3 0.1280910.128091 \infty
(1:1:1)+(0:1:1)(1:1:0)(1:0:0)(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0) x(x+z)x (x + z) == 0,0, yy == 2xz2z3-2xz^2 - z^3 0.1280910.128091 \infty
Generator D0D_0 Height Order
(0:1:1)(1:1:0)(0 : -1 : 1) - (1 : -1 : 0) zxz x == 0,0, yy == z3-z^3 0.1280910.128091 \infty
(1:1:1)+(0:1:1)(1:1:0)(1:0:0)(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0) x(x+z)x (x + z) == 0,0, yy == 2xz2z3-2xz^2 - z^3 0.1280910.128091 \infty
Generator D0D_0 Height Order
(0:1:1)(1:1:0)(0 : -1 : 1) - (1 : -1 : 0) zxz x == 0,0, yy == x3+xz2z3x^3 + xz^2 - z^3 0.1280910.128091 \infty
(1:1:1)+(0:1:1)(1:1:0)(1:1:0)(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0) x(x+z)x (x + z) == 0,0, yy == x33xz2z3x^3 - 3xz^2 - z^3 0.1280910.128091 \infty

2-torsion field: 6.0.2728448.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: 22
Mordell-Weil rank: 22
2-Selmer rank:22
Regulator: 0.012305 0.012305
Real period: 20.94165 20.94165
Tamagawa product: 3 3
Torsion order:1 1
Leading coefficient: 0.773095 0.773095
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 33 33 1+T+T21 + T + T^{2}
7373 22 22 11 1+17T+73T21 + 17 T + 73 T^{2}

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 2.40.3 no
33 3.480.12 no

Sato-Tate group

ST\mathrm{ST}\simeq E6E_6
ST0\mathrm{ST}^0\simeq SU(2)\mathrm{SU}(2)

Decomposition of the Jacobian

Splits over the number field Q(b)\Q (b) \simeq 6.6.2073071593.1 with defining polynomial:
  x6x530x4+31x3+206x2150x81x^{6} - x^{5} - 30 x^{4} + 31 x^{3} + 206 x^{2} - 150 x - 81

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=129991324b52909027b4+37400354b3+4820093324b21726571108b307474g_4 = -\frac{129991}{324} b^{5} - \frac{29090}{27} b^{4} + \frac{374003}{54} b^{3} + \frac{4820093}{324} b^{2} - \frac{1726571}{108} b - \frac{30747}{4}
  g6=5632890972916b5150339193243b4+18710389154b3+237837713512916b28144724245972b14429566936g_6 = -\frac{563289097}{2916} b^{5} - \frac{150339193}{243} b^{4} + \frac{187103891}{54} b^{3} + \frac{23783771351}{2916} b^{2} - \frac{8144724245}{972} b - \frac{144295669}{36}
   Conductor norm: 64

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+32]\Z [\frac{1 + \sqrt{-3}}{2}]
End(J)Q\End (J_{}) \otimes \Q \simeqQ(3)\Q(\sqrt{-3})
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 6.6.2073071593.1 with defining polynomial x6x530x4+31x3+206x2150x81x^{6} - x^{5} - 30 x^{4} + 31 x^{3} + 206 x^{2} - 150 x - 81

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

Endomorphism ring over Q\overline{\Q}:

End(JQ)\End (J_{\overline{\Q}})\simeqan Eichler order of index 33 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(73)\Q(\sqrt{73}) with generator 127a5427a4+1927a3+7327a2229a5-\frac{1}{27} a^{5} - \frac{4}{27} a^{4} + \frac{19}{27} a^{3} + \frac{73}{27} a^{2} - \frac{22}{9} a - 5 with minimal polynomial x2x18x^{2} - x - 18:

End(JF)\End (J_{F})\simeqZ[1+32]\Z [\frac{1 + \sqrt{-3}}{2}]
End(JF)Q\End (J_{F}) \otimes \Q \simeqQ(3)\Q(\sqrt{-3})
End(JF)R\End (J_{F}) \otimes \R\simeq C\C
  Sato Tate group: E3E_3
  Of GL2\GL_2-type, simple

Over subfield FF \simeq 3.3.5329.1 with generator 136a5+19a4518a35536a23512a+14\frac{1}{36} a^{5} + \frac{1}{9} a^{4} - \frac{5}{18} a^{3} - \frac{55}{36} a^{2} - \frac{35}{12} a + \frac{1}{4} with minimal polynomial x3x224x+27x^{3} - x^{2} - 24 x + 27:

End(JF)\End (J_{F})\simeqZ[1+32]\Z [\frac{1 + \sqrt{-3}}{2}]
End(JF)Q\End (J_{F}) \otimes \Q \simeqQ(3)\Q(\sqrt{-3})
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);