Properties

Label 139968.b.839808.1
Conductor $139968$
Discriminant $-839808$
Mordell-Weil group \(\Z \oplus \Z/{3}\Z\)
Sato-Tate group $J(E_6)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = x^3 + 3$ (homogenize, simplify)
$y^2 + x^3y = x^3z^3 + 3z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^3 + 12$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 0, 0, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 0, 0, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([12, 0, 0, 4, 0, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(139968\) \(=\) \( 2^{6} \cdot 3^{7} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-839808\) \(=\) \( - 2^{7} \cdot 3^{8} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(116\) \(=\)  \( 2^{2} \cdot 29 \)
\( I_4 \)  \(=\) \(513\) \(=\)  \( 3^{3} \cdot 19 \)
\( I_6 \)  \(=\) \(16623\) \(=\)  \( 3^{2} \cdot 1847 \)
\( I_{10} \)  \(=\) \(432\) \(=\)  \( 2^{4} \cdot 3^{3} \)
\( J_2 \)  \(=\) \(348\) \(=\)  \( 2^{2} \cdot 3 \cdot 29 \)
\( J_4 \)  \(=\) \(1968\) \(=\)  \( 2^{4} \cdot 3 \cdot 41 \)
\( J_6 \)  \(=\) \(-3856\) \(=\)  \( - 2^{4} \cdot 241 \)
\( J_8 \)  \(=\) \(-1303728\) \(=\)  \( - 2^{4} \cdot 3 \cdot 157 \cdot 173 \)
\( J_{10} \)  \(=\) \(839808\) \(=\)  \( 2^{7} \cdot 3^{8} \)
\( g_1 \)  \(=\) \(164089192/27\)
\( g_2 \)  \(=\) \(7999592/81\)
\( g_3 \)  \(=\) \(-405362/729\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -1 : 1),\, (-1 : 2 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -3 : 1),\, (-1 : 3 : 1)\)

magma: [C![-1,-1,1],C![-1,2,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-1,-3,1],C![-1,3,1],C![1,-1,0],C![1,1,0]]; // simplified model
 

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: \(\Z \oplus \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.364885\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.364885\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)
Generator $D_0$ Height Order
\((-1 : -3 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2z^3\) \(0.364885\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0\) \(3\)

2-torsion field: 6.0.1492992.5

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.364885 \)
Real period: \( 10.97281 \)
Tamagawa product: \( 6 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.669217 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(7\) \(3\) \(1\)
\(3\) \(7\) \(8\) \(2\) \(1\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.20.3 no
\(3\) 3.960.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.2.45349632.2 with defining polynomial:
  \(x^{6} - 12\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 72 b^{5} + 540 b^{2}\)
  \(g_6 = -9072 b^{3} - 45360\)
   Conductor norm: 9
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -72 b^{5} + 540 b^{2}\)
  \(g_6 = 9072 b^{3} - 45360\)
   Conductor norm: 9

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} - 3 x^{10} - 30 x^{8} + 77 x^{6} + 510 x^{4} + 537 x^{2} + 169\)

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 \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{78}{6137} a^{10} + \frac{225}{6137} a^{8} + \frac{2602}{6137} a^{6} - \frac{7122}{6137} a^{4} - \frac{42018}{6137} a^{2} - \frac{20770}{6137}\) with minimal polynomial \(x^{2} - x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_6$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{3}) \) with generator \(\frac{411}{9386} a^{11} - \frac{1597}{9386} a^{9} - \frac{10653}{9386} a^{7} + \frac{39369}{9386} a^{5} + \frac{172859}{9386} a^{3} + \frac{91617}{9386} a\) with minimal polynomial \(x^{2} - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-\frac{27}{159562} a^{11} - \frac{829}{159562} a^{9} + \frac{5373}{159562} a^{7} + \frac{19449}{159562} a^{5} - \frac{129483}{159562} a^{3} - \frac{283131}{159562} a\) with minimal polynomial \(x^{2} + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.972.1 with generator \(\frac{1}{361} a^{10} - \frac{8}{361} a^{8} + \frac{10}{361} a^{6} + \frac{27}{361} a^{4} + \frac{14}{361} a^{2} + \frac{828}{361}\) with minimal polynomial \(x^{3} - 12\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.972.1 with generator \(-\frac{281}{6137} a^{10} + \frac{1146}{6137} a^{8} + \frac{6823}{6137} a^{6} - \frac{26701}{6137} a^{4} - \frac{109954}{6137} a^{2} - \frac{66361}{6137}\) with minimal polynomial \(x^{3} - 12\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.972.1 with generator \(\frac{264}{6137} a^{10} - \frac{1010}{6137} a^{8} - \frac{6993}{6137} a^{6} + \frac{26242}{6137} a^{4} + \frac{109716}{6137} a^{2} + \frac{52285}{6137}\) with minimal polynomial \(x^{3} - 12\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\zeta_{12})\) with generator \(\frac{1740}{79781} a^{11} - \frac{13989}{159562} a^{9} - \frac{43932}{79781} a^{7} + \frac{344361}{159562} a^{5} + \frac{702280}{79781} a^{3} + \frac{637179}{159562} a\) with minimal polynomial \(x^{4} - x^{2} + 1\):

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

Over subfield \(F \simeq \) 6.0.2834352.1 with generator \(-\frac{43}{6137} a^{10} + \frac{99}{12274} a^{8} + \frac{1774}{6137} a^{6} - \frac{5343}{12274} a^{4} - \frac{31686}{6137} a^{2} - \frac{54165}{12274}\) with minimal polynomial \(x^{6} + 12\):

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

Over subfield \(F \simeq \) 6.0.15116544.4 with generator \(\frac{21}{159562} a^{11} - \frac{1}{722} a^{10} - \frac{844}{79781} a^{9} + \frac{4}{361} a^{8} + \frac{4209}{79781} a^{7} - \frac{5}{361} a^{6} + \frac{11332}{79781} a^{5} - \frac{27}{722} a^{4} - \frac{151231}{159562} a^{3} - \frac{7}{361} a^{2} - \frac{325657}{159562} a - \frac{414}{361}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 6.2.45349632.2 with generator \(-\frac{3507}{159562} a^{11} + \frac{6580}{79781} a^{9} + \frac{93237}{159562} a^{7} - \frac{162456}{79781} a^{5} - \frac{1534043}{159562} a^{3} - \frac{539936}{79781} a\) with minimal polynomial \(x^{6} - 12\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.45349632.2 with generator \(-\frac{2001}{159562} a^{11} + \frac{6679}{159562} a^{9} + \frac{62279}{159562} a^{7} - \frac{204621}{159562} a^{5} - \frac{962985}{159562} a^{3} - \frac{256889}{159562} a\) with minimal polynomial \(x^{6} - 12\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.45349632.2 with generator \(\frac{162}{4693} a^{11} - \frac{1167}{9386} a^{9} - \frac{4574}{4693} a^{7} + \frac{31149}{9386} a^{5} + \frac{73442}{4693} a^{3} + \frac{78633}{9386} a\) with minimal polynomial \(x^{6} - 12\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.15116544.4 with generator \(\frac{3313}{159562} a^{11} + \frac{281}{12274} a^{10} - \frac{10563}{159562} a^{9} - \frac{573}{6137} a^{8} - \frac{50410}{79781} a^{7} - \frac{6823}{12274} a^{6} + \frac{149748}{79781} a^{5} + \frac{26701}{12274} a^{4} + \frac{830216}{79781} a^{3} + \frac{54977}{6137} a^{2} + \frac{1020453}{159562} a + \frac{66361}{12274}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 6.0.15116544.4 with generator \(-\frac{1667}{79781} a^{11} - \frac{132}{6137} a^{10} + \frac{12251}{159562} a^{9} + \frac{505}{6137} a^{8} + \frac{46201}{79781} a^{7} + \frac{6993}{12274} a^{6} - \frac{161080}{79781} a^{5} - \frac{13121}{6137} a^{4} - \frac{1509201}{159562} a^{3} - \frac{54858}{6137} a^{2} - \frac{347398}{79781} a - \frac{52285}{12274}\) with minimal polynomial \(x^{6} - 6 x^{3} + 18\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);