Properties

Label 8649.b.700569.1
Conductor $8649$
Discriminant $700569$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = 9x^5 + 2x^4 - 21x^3 - 22x^2 - 8x - 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 9x^5z + 2x^4z^2 - 21x^3z^3 - 22x^2z^4 - 8xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 36x^5 + 9x^4 - 82x^3 - 87x^2 - 32x - 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -8, -22, -21, 2, 9]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -8, -22, -21, 2, 9], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-4, -32, -87, -82, 9, 36]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8649\) \(=\) \( 3^{2} \cdot 31^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(700569\) \(=\) \( 3^{6} \cdot 31^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1132\) \(=\)  \( 2^{2} \cdot 283 \)
\( I_4 \)  \(=\) \(73377\) \(=\)  \( 3^{2} \cdot 31 \cdot 263 \)
\( I_6 \)  \(=\) \(21088959\) \(=\)  \( 3 \cdot 17 \cdot 31 \cdot 13339 \)
\( I_{10} \)  \(=\) \(369024\) \(=\)  \( 2^{7} \cdot 3 \cdot 31^{2} \)
\( J_2 \)  \(=\) \(849\) \(=\)  \( 3 \cdot 283 \)
\( J_4 \)  \(=\) \(2517\) \(=\)  \( 3 \cdot 839 \)
\( J_6 \)  \(=\) \(-2507\) \(=\)  \( - 23 \cdot 109 \)
\( J_8 \)  \(=\) \(-2115933\) \(=\)  \( - 3 \cdot 823 \cdot 857 \)
\( J_{10} \)  \(=\) \(700569\) \(=\)  \( 3^{6} \cdot 31^{2} \)
\( g_1 \)  \(=\) \(1815232161643/2883\)
\( g_2 \)  \(=\) \(19016091893/8649\)
\( g_3 \)  \(=\) \(-200783123/77841\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_6$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![-2,3,3],C![-1,3,3],C![1,0,0]];
 

Number of rational Weierstrass points: \(3\)

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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 3 : 3) + (-1 : 3 : 3) - 2 \cdot(1 : 0 : 0)\) \((3x + z) (3x + 2z)\) \(=\) \(0,\) \(9y\) \(=\) \(z^3\) \(0\) \(2\)
\((-1 : 3 : 3) - (1 : 0 : 0)\) \(3x + z\) \(=\) \(0,\) \(9y\) \(=\) \(z^3\) \(0\) \(2\)

2-torsion field: 3.3.961.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 5.541099 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.385274 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(6\) \(4\) \(1 - 3 T + 3 T^{2}\)
\(31\) \(2\) \(2\) \(1\) \(1 - 4 T + 31 T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_6$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.772987077.1 with defining polynomial:
  \(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{16934823333}{3104} b^{5} + \frac{15604267599}{1552} b^{4} + \frac{223933318713}{1552} b^{3} - \frac{1241168535621}{3104} b^{2} - \frac{111941475201}{1552} b + \frac{462072237363}{776}\)
  \(g_6 = -\frac{579872195059509}{6208} b^{5} + \frac{1068664047901587}{6208} b^{4} + \frac{15335612391671001}{6208} b^{3} - \frac{2656270916385201}{388} b^{2} - \frac{958200176670183}{776} b + \frac{7911131565422907}{776}\)
   Conductor norm: 1

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.772987077.1 with defining polynomial \(x^{6} - x^{5} - 28 x^{4} + 51 x^{3} + 75 x^{2} - 98 x - 92\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{93}) \) with generator \(-\frac{1}{2} a^{4} + 13 a^{2} - \frac{23}{2} a - 23\) with minimal polynomial \(x^{2} - x - 23\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.961.1 with generator \(-\frac{17}{97} a^{5} + \frac{1}{97} a^{4} + \frac{437}{97} a^{3} - \frac{450}{97} a^{2} - \frac{700}{97} a + \frac{60}{97}\) with minimal polynomial \(x^{3} - x^{2} - 10 x + 8\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple