Properties

Label 8281.b.405769.1
Conductor $8281$
Discriminant $405769$
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + (x^3 + x + 1)y = -3x^5 + 9x^4 - 7x^3 - 2x^2 + x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -3x^5z + 9x^4z^2 - 7x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 12x^5 + 38x^4 - 26x^3 - 7x^2 + 6x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, -7, 9, -3]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, -7, 9, -3], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, -7, -26, 38, -12, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8281\) \(=\) \( 7^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(405769\) \(=\) \( 7^{4} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2596\) \(=\)  \( 2^{2} \cdot 11 \cdot 59 \)
\( I_4 \)  \(=\) \(375193\) \(=\)  \( 7^{2} \cdot 13 \cdot 19 \cdot 31 \)
\( I_6 \)  \(=\) \(248614093\) \(=\)  \( 7^{2} \cdot 13 \cdot 390289 \)
\( I_{10} \)  \(=\) \(51938432\) \(=\)  \( 2^{7} \cdot 7^{4} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(649\) \(=\)  \( 11 \cdot 59 \)
\( J_4 \)  \(=\) \(1917\) \(=\)  \( 3^{3} \cdot 71 \)
\( J_6 \)  \(=\) \(-1907\) \(=\)  \( -1907 \)
\( J_8 \)  \(=\) \(-1228133\) \(=\)  \( -1228133 \)
\( J_{10} \)  \(=\) \(405769\) \(=\)  \( 7^{4} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(115139273278249/405769\)
\( g_2 \)  \(=\) \(524030063733/405769\)
\( g_3 \)  \(=\) \(-803230307/405769\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -2 : 1)\)
\((1 : -1 : 3)\) \((3 : -6 : 2)\) \((-2 : -13 : 1)\) \((-2 : 22 : 1)\) \((1 : -36 : 3)\) \((3 : -41 : 2)\)

magma: [C![-2,-13,1],C![-2,22,1],C![0,-1,1],C![0,0,1],C![1,-36,3],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,-1,3],C![1,0,0],C![3,-41,2],C![3,-6,2]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 4xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-6xz^2 + 3z^3\) \(0.086938\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-4xz^2 + z^3\) \(0.086938\) \(\infty\)

2-torsion field: 9.9.567869252041.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.005668 \)
Real period: \( 19.78540 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.336475 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(2\) \(4\) \(3\) \(1 + 5 T + 7 T^{2}\)
\(13\) \(2\) \(2\) \(1\) \(1 + 2 T + 13 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.891474493.2 with defining polynomial:
  \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{1574069}{6561} b^{5} + \frac{13576892}{6561} b^{4} - \frac{56218393}{6561} b^{3} - \frac{17230402}{243} b^{2} + \frac{2496578}{243} b + \frac{81063052}{243}\)
  \(g_6 = -\frac{655459038871}{531441} b^{5} + \frac{100362384668}{531441} b^{4} + \frac{20455326360833}{531441} b^{3} + \frac{543846776200}{19683} b^{2} - \frac{3528969869152}{19683} b - \frac{1057160135213}{19683}\)
   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 \(K = \Q (a) \simeq \) 6.6.891474493.2 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 162 x^{2} - 81 x - 27\)

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 \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(\frac{4}{243} a^{5} - \frac{1}{243} a^{4} - \frac{145}{243} a^{3} - \frac{32}{243} a^{2} + \frac{316}{81} a - \frac{11}{27}\) with minimal polynomial \(x^{2} - x - 3\):

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

Over subfield \(F \simeq \) 3.3.8281.2 with generator \(-\frac{1}{27} a^{5} - \frac{2}{27} a^{4} + \frac{34}{27} a^{3} + \frac{71}{27} a^{2} - \frac{49}{9} a - \frac{7}{3}\) with minimal polynomial \(x^{3} - x^{2} - 30 x - 27\):

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