Properties

Label 13689.a.13689.1
Conductor $13689$
Discriminant $13689$
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

$y^2 + (x^3 + x + 1)y = x^5 - x^4 - 4x^3 - 2x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - x^4z^2 - 4x^3z^3 - 2x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 2x^4 - 14x^3 - 7x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, -4, -1, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, -4, -1, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -7, -14, -2, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(13689\) \(=\) \( 3^{4} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(13689\) \(=\) \( 3^{4} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(516\) \(=\)  \( 2^{2} \cdot 3 \cdot 43 \)
\( I_4 \)  \(=\) \(8073\) \(=\)  \( 3^{3} \cdot 13 \cdot 23 \)
\( I_6 \)  \(=\) \(1250613\) \(=\)  \( 3^{3} \cdot 7 \cdot 13 \cdot 509 \)
\( I_{10} \)  \(=\) \(1752192\) \(=\)  \( 2^{7} \cdot 3^{4} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(129\) \(=\)  \( 3 \cdot 43 \)
\( J_4 \)  \(=\) \(357\) \(=\)  \( 3 \cdot 7 \cdot 17 \)
\( J_6 \)  \(=\) \(-347\) \(=\)  \( -347 \)
\( J_8 \)  \(=\) \(-43053\) \(=\)  \( - 3 \cdot 113 \cdot 127 \)
\( J_{10} \)  \(=\) \(13689\) \(=\)  \( 3^{4} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(441025329/169\)
\( g_2 \)  \(=\) \(9461333/169\)
\( g_3 \)  \(=\) \(-641603/1521\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((2 : -3 : 1)\) \((-1 : -6 : 3)\) \((2 : -8 : 1)\) \((-1 : -11 : 3)\) \((-3 : 13 : 2)\) \((-3 : 18 : 2)\)

magma: [C![-3,13,2],C![-3,18,2],C![-1,-11,3],C![-1,-6,3],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0],C![2,-8,1],C![2,-3,1]];
 

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
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.150903\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.150903\) \(\infty\)

2-torsion field: 9.9.2565164201769.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.017078 \)
Real period: \( 22.36905 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.382038 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.2436053373.2 with defining polynomial:
  \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{12015765}{3956} b^{5} + \frac{233936235}{15824} b^{4} + \frac{735217587}{15824} b^{3} - \frac{2326501125}{15824} b^{2} - \frac{240909201}{688} b - \frac{288784197}{3956}\)
  \(g_6 = \frac{36685451205}{3956} b^{5} - \frac{2858709285549}{63296} b^{4} - \frac{8968980313215}{63296} b^{3} + \frac{28420667571231}{63296} b^{2} + \frac{2939506491519}{2752} b + \frac{14100757396245}{63296}\)
   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 \(K = \Q (a) \simeq \) 6.6.2436053373.2 with defining polynomial \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

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 \(-\frac{9}{989} a^{5} - \frac{12}{989} a^{4} + \frac{335}{989} a^{3} + \frac{351}{989} a^{2} - \frac{117}{43} a - \frac{1930}{989}\) 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 \(F \simeq \) 3.3.13689.2 with generator \(\frac{4}{129} a^{5} - \frac{3}{43} a^{4} - \frac{56}{43} a^{3} + \frac{145}{129} a^{2} + \frac{585}{43} a + \frac{442}{43}\) with minimal polynomial \(x^{3} - 39 x - 91\):

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