Properties

Label 20736.i.373248.1
Conductor $20736$
Discriminant $373248$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $D_{2,1}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathsf{CM})\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(40\) \(=\)  \( 2^{3} \cdot 5 \)
\( I_4 \)  \(=\) \(45\) \(=\)  \( 3^{2} \cdot 5 \)
\( I_6 \)  \(=\) \(555\) \(=\)  \( 3 \cdot 5 \cdot 37 \)
\( I_{10} \)  \(=\) \(6\) \(=\)  \( 2 \cdot 3 \)
\( J_2 \)  \(=\) \(240\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(1320\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
\( J_6 \)  \(=\) \(-2560\) \(=\)  \( - 2^{9} \cdot 5 \)
\( J_8 \)  \(=\) \(-589200\) \(=\)  \( - 2^{4} \cdot 3 \cdot 5^{2} \cdot 491 \)
\( J_{10} \)  \(=\) \(373248\) \(=\)  \( 2^{9} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(6400000/3\)
\( g_2 \)  \(=\) \(440000/9\)
\( g_3 \)  \(=\) \(-32000/81\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_3:D_4$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 7.223966 \)
Tamagawa product: \( 6 \)
Torsion order:\( 6 \)
Leading coefficient: \( 1.203994 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(9\) \(3\) \(1\)
\(3\) \(4\) \(6\) \(2\) \(1\)

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?
\(2\) 2.180.4 yes
\(3\) 3.8640.8 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $D_{2,1}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 576.f
  Elliptic curve isogeny class 36.a

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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 non-Eichler order of index \(16\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-3}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

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

\(\End (J_{F})\)\(\simeq\)an order of index \(4\) in \(\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
  Sato Tate group: $C_2$
  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 \(x^{2} - 6\):

\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(12\) 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

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);