Properties

Label 15552.c.746496.1
Conductor $15552$
Discriminant $746496$
Mordell-Weil group \(\Z/{6}\Z\)
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

$y^2 + x^3y = 3x^3 + 8$ (homogenize, simplify)
$y^2 + x^3y = 3x^3z^3 + 8z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 12x^3 + 32$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 0, 0, 3]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![8, 0, 0, 3], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([32, 0, 0, 12, 0, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(15552\) \(=\) \( 2^{6} \cdot 3^{5} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(746496\) \(=\) \( 2^{10} \cdot 3^{6} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(142\) \(=\)  \( 2 \cdot 71 \)
\( I_4 \)  \(=\) \(1368\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 19 \)
\( I_6 \)  \(=\) \(47940\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 47 \)
\( I_{10} \)  \(=\) \(-12\) \(=\)  \( - 2^{2} \cdot 3 \)
\( J_2 \)  \(=\) \(852\) \(=\)  \( 2^{2} \cdot 3 \cdot 71 \)
\( J_4 \)  \(=\) \(-2586\) \(=\)  \( - 2 \cdot 3 \cdot 431 \)
\( J_6 \)  \(=\) \(-2596\) \(=\)  \( - 2^{2} \cdot 11 \cdot 59 \)
\( J_8 \)  \(=\) \(-2224797\) \(=\)  \( - 3 \cdot 741599 \)
\( J_{10} \)  \(=\) \(-746496\) \(=\)  \( - 2^{10} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(-1804229351/3\)
\( g_2 \)  \(=\) \(154259641/72\)
\( g_3 \)  \(=\) \(3271609/1296\)

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

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

Number of rational Weierstrass points: \(1\)

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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 3.1.108.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 8.616344 \)
Tamagawa product: \( 4 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.957371 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(10\) \(2\) \(1 + 2 T^{2}\)
\(3\) \(5\) \(6\) \(2\) \(1\)

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 \(\Q (b) \simeq \) 6.2.1492992.4 with defining polynomial:
  \(x^{6} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 2880 b^{4} - 3456 b\)
  \(g_6 = 290304 b^{3} - 366336\)
   Conductor norm: 81
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 2880 b^{4} + 3456 b\)
  \(g_6 = -290304 b^{3} - 366336\)
   Conductor norm: 81

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 \(x^{12} - 6 x^{10} + 44 x^{6} + 60 x^{4} + 24 x^{2} + 4\)

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 \(-\frac{1}{10} a^{10} + \frac{11}{20} a^{8} + \frac{2}{5} a^{6} - \frac{26}{5} a^{4} - \frac{71}{10} a^{2} - \frac{6}{5}\) 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 \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(-\frac{12}{31} a^{11} + \frac{145}{62} a^{9} + \frac{3}{62} a^{7} - \frac{561}{31} a^{5} - \frac{654}{31} a^{3} - \frac{129}{31} a\) 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: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-6}) \) with generator \(\frac{49}{310} a^{11} - \frac{287}{310} a^{9} - \frac{41}{310} a^{7} + \frac{1064}{155} a^{5} + \frac{1622}{155} a^{3} + \frac{1019}{155} a\) 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: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(-\frac{26}{155} a^{10} + \frac{371}{310} a^{8} - \frac{206}{155} a^{6} - \frac{952}{155} a^{4} - \frac{456}{155} a^{2} + \frac{108}{155}\) with minimal polynomial \(x^{3} - 2\):

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

Over subfield \(F \simeq \) 3.1.108.1 with generator \(\frac{159}{620} a^{10} - \frac{521}{310} a^{8} + \frac{151}{155} a^{6} + \frac{3209}{310} a^{4} + \frac{1651}{155} a^{2} + \frac{282}{155}\) with minimal polynomial \(x^{3} - 2\):

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

Over subfield \(F \simeq \) 3.1.108.1 with generator \(-\frac{11}{124} a^{10} + \frac{15}{31} a^{8} + \frac{11}{31} a^{6} - \frac{261}{62} a^{4} - \frac{239}{31} a^{2} - \frac{78}{31}\) with minimal polynomial \(x^{3} - 2\):

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

Over subfield \(F \simeq \) \(\Q(\sqrt{2}, \sqrt{-3})\) with generator \(\frac{71}{620} a^{11} - \frac{219}{310} a^{9} + \frac{13}{310} a^{7} + \frac{1741}{310} a^{5} + \frac{824}{155} a^{3} - \frac{187}{155} a\) with minimal polynomial \(x^{4} + 2 x^{2} + 4\):

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

Over subfield \(F \simeq \) 6.0.34992.1 with generator \(\frac{9}{620} a^{10} - \frac{97}{620} a^{8} + \frac{137}{310} a^{6} + \frac{129}{310} a^{4} - \frac{553}{310} a^{2} - \frac{218}{155}\) with minimal polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\):

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

Over subfield \(F \simeq \) 6.0.4478976.2 with generator \(\frac{23}{310} a^{11} - \frac{203}{620} a^{9} - \frac{247}{310} a^{7} + \frac{588}{155} a^{5} + \frac{2943}{310} a^{3} + \frac{763}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):

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

Over subfield \(F \simeq \) 6.2.1492992.4 with generator \(\frac{123}{620} a^{11} - \frac{809}{620} a^{9} + \frac{219}{310} a^{7} + \frac{2693}{310} a^{5} + \frac{1949}{310} a^{3} - \frac{86}{155} a\) with minimal polynomial \(x^{6} - 2\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.2.1492992.4 with generator \(\frac{15}{62} a^{11} - \frac{189}{124} a^{9} + \frac{33}{62} a^{7} + \frac{308}{31} a^{5} + \frac{771}{62} a^{3} + \frac{131}{31} a\) with minimal polynomial \(x^{6} - 2\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.2.1492992.4 with generator \(\frac{273}{620} a^{11} - \frac{877}{310} a^{9} + \frac{192}{155} a^{7} + \frac{5773}{310} a^{5} + \frac{2902}{155} a^{3} + \frac{569}{155} a\) with minimal polynomial \(x^{6} - 2\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.0.4478976.2 with generator \(\frac{61}{620} a^{11} - \frac{117}{155} a^{9} + \frac{343}{310} a^{7} + \frac{1081}{310} a^{5} - \frac{126}{155} a^{3} - \frac{272}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):

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

Over subfield \(F \simeq \) 6.0.4478976.2 with generator \(\frac{9}{620} a^{11} - \frac{97}{620} a^{9} + \frac{137}{310} a^{7} + \frac{129}{310} a^{5} - \frac{553}{310} a^{3} - \frac{528}{155} a\) with minimal polynomial \(x^{6} + 6 x^{4} + 12 x^{2} + 6\):

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