Properties

Label 20736.l.373248.1
Conductor 20736
Discriminant 373248
Mordell-Weil group \(\Z/{2}\Z \times \Z/{6}\Z\)
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{QM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, -1, 9, 6], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, -1, 9, 6]), R([1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, -1, 9, 6], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, -12, -4, 36, 24]))
 

$y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$ (homogenize, simplify)
$y^2 + z^3y = 6x^5z + 9x^4z^2 - x^3z^3 - 3x^2z^4$ (dehomogenize, simplify)
$y^2 = 24x^5 + 36x^4 - 4x^3 - 12x^2 + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(20736\) = \( 2^{8} \cdot 3^{4} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(373248\) = \( 2^{9} \cdot 3^{6} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(7008\) =  \( 2^{5} \cdot 3 \cdot 73 \)
\( I_4 \)  = \(1700352\) =  \( 2^{9} \cdot 3^{4} \cdot 41 \)
\( I_6 \)  = \(3259367424\) =  \( 2^{17} \cdot 3^{4} \cdot 307 \)
\( I_{10} \)  = \(1528823808\) =  \( 2^{21} \cdot 3^{6} \)
\( J_2 \)  = \(876\) =  \( 2^{2} \cdot 3 \cdot 73 \)
\( J_4 \)  = \(14262\) =  \( 2 \cdot 3 \cdot 2377 \)
\( J_6 \)  = \(207364\) =  \( 2^{2} \cdot 47 \cdot 1103 \)
\( J_8 \)  = \(-5438445\) =  \( - 3 \cdot 5 \cdot 37 \cdot 41 \cdot 239 \)
\( J_{10} \)  = \(373248\) =  \( 2^{9} \cdot 3^{6} \)
\( g_1 \)  = \(4146143186/3\)
\( g_2 \)  = \(924693409/36\)
\( g_3 \)  = \(276260689/648\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![-1,-4,2],C![0,-1,1],C![0,0,1],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (-1 : -4 : 2)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(2\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{2}\Z \times \Z/{6}\Z\)

Generator Height Order
\(2x + z\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
\(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{3})\)

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 18.60015 \)
Tamagawa product: \( 8 \)
Torsion order:\( 12 \)
Leading coefficient: \( 1.033342 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(9\) \(8\) \(2\) \(1 + 2 T^{2}\)
\(3\) \(6\) \(4\) \(4\) \(1\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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 \(K = \Q (a) \simeq \) 8.0.339738624.10 with defining polynomial \(x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(6\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{15} a^{6} - \frac{1}{6} a^{4} - \frac{2}{3} a^{2} - \frac{3}{5}\) with minimal polynomial \(x^{2} - x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [3\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_4$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-\frac{1}{10} a^{6} - \frac{2}{5}\) with minimal polynomial \(x^{2} + 2\):

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

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(-\frac{1}{6} a^{6} - \frac{2}{3} a^{4} - \frac{2}{3} a^{2} - 2\) with minimal polynomial \(x^{2} - 6\):

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

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{-3})\) with generator \(-\frac{2}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{6}{5}\) with minimal polynomial \(x^{4} - 2 x^{2} + 4\):

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

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(\frac{2}{45} a^{7} - \frac{1}{18} a^{5} + \frac{1}{9} a^{3} + \frac{11}{15} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{7}{90} a^{7} + \frac{5}{18} a^{5} + \frac{4}{9} a^{3} + \frac{23}{15} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

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

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

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

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(-\frac{1}{15} a^{7} - \frac{1}{6} a^{5} - \frac{2}{3} a^{3} - \frac{3}{5} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple