Properties

Label 20736.k.373248.1
Conductor 20736
Discriminant -373248
Mordell-Weil group \(\Z \times \Z/{6}\Z\)
Sato-Tate group $D_{2,1}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathrm{CM})\)
\(\overline{\Q}\)-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)
$y^2 = x^6 + 8$ (minimize, homogenize)

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

\( I_2 \)  \(=\) \(-1920\) \(=\)  \( - 2^{7} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(103680\) \(=\)  \( 2^{8} \cdot 3^{4} \cdot 5 \)
\( I_6 \)  \(=\) \(-61378560\) \(=\)  \( - 2^{12} \cdot 3^{4} \cdot 5 \cdot 37 \)
\( I_{10} \)  \(=\) \(-1528823808\) \(=\)  \( - 2^{21} \cdot 3^{6} \)
\( J_2 \)  \(=\) \(-240\) \(=\)  \( - 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(1320\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
\( J_6 \)  \(=\) \(2560\) \(=\)  \( 2^{9} \cdot 5 \)
\( J_8 \)  \(=\) \(-589200\) \(=\)  \( - 2^{4} \cdot 3 \cdot 5^{2} \cdot 491 \)
\( J_{10} \)  \(=\) \(-373248\) \(=\)  \( - 2^{9} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(6400000/3\)
\( g_2 \)  \(=\) \(440000/9\)
\( g_3 \)  \(=\) \(-32000/81\)

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

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.326617\) \(\infty\)
\((-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0\) \(6\)

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.326617 \)
Real period: \( 12.51227 \)
Tamagawa product: \( 12 \)
Torsion order:\( 6 \)
Leading coefficient: \( 1.362242 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 curves:
  Elliptic curve 576.e4
  Elliptic curve 36.a4

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 \(F \simeq \) \(\Q(\sqrt{-6}) \) 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

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