Properties

Label 5184.a.46656.1
Conductor $5184$
Discriminant $-46656$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $J(C_2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathsf{CM})\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = x^3 + 2$ (homogenize, simplify)
$y^2 + x^3y = x^3z^3 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^3 + 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 0, 0, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 0, 0, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([8, 0, 0, 4, 0, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(5184\) \(=\) \( 2^{6} \cdot 3^{4} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-46656\) \(=\) \( - 2^{6} \cdot 3^{6} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(76\) \(=\)  \( 2^{2} \cdot 19 \)
\( I_4 \)  \(=\) \(252\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 7 \)
\( I_6 \)  \(=\) \(5160\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \cdot 43 \)
\( I_{10} \)  \(=\) \(24\) \(=\)  \( 2^{3} \cdot 3 \)
\( J_2 \)  \(=\) \(228\) \(=\)  \( 2^{2} \cdot 3 \cdot 19 \)
\( J_4 \)  \(=\) \(654\) \(=\)  \( 2 \cdot 3 \cdot 109 \)
\( J_6 \)  \(=\) \(-644\) \(=\)  \( - 2^{2} \cdot 7 \cdot 23 \)
\( J_8 \)  \(=\) \(-143637\) \(=\)  \( - 3 \cdot 13 \cdot 29 \cdot 127 \)
\( J_{10} \)  \(=\) \(46656\) \(=\)  \( 2^{6} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(39617584/3\)
\( g_2 \)  \(=\) \(1495262/9\)
\( g_3 \)  \(=\) \(-58121/81\)

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

magma: [C![1,-1,0],C![1,0,0]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: \(\Q(\zeta_{12})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 14.11020 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.783900 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(C_2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 2.2.8.1-81.1-b3

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{-2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-2}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}, \sqrt{-3})\) with defining polynomial \(x^{4} + 2 x^{2} + 4\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-6}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

Remainder of the endomorphism lattice by field

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

\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(8\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
  Sato Tate group: $C_{2,1}$
  Not of \(\GL_2\)-type, not simple

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

\(\End (J_{F})\)\(\simeq\)an order of conductor of norm \(2\) in \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-2}, \sqrt{3})\) (CM)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, simple

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

\(\End (J_{F})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)the quaternion algebra over \(\Q\) of discriminant 2
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\H\)
  Sato Tate group: $J(C_1)$
  Not of \(\GL_2\)-type, simple