Properties

Label 26244.e.472392.1
Conductor $26244$
Discriminant $-472392$
Mordell-Weil group \(\Z/{3}\Z \times \Z/{3}\Z\)
Sato-Tate group $J(E_3)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^3 + 9$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, 0, 0, 2, 0, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(26244\) \(=\) \( 2^{2} \cdot 3^{8} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-472392\) \(=\) \( - 2^{3} \cdot 3^{10} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(356\) \(=\)  \( 2^{2} \cdot 89 \)
\( I_4 \)  \(=\) \(3969\) \(=\)  \( 3^{4} \cdot 7^{2} \)
\( I_6 \)  \(=\) \(419553\) \(=\)  \( 3^{3} \cdot 41 \cdot 379 \)
\( I_{10} \)  \(=\) \(248832\) \(=\)  \( 2^{10} \cdot 3^{5} \)
\( J_2 \)  \(=\) \(267\) \(=\)  \( 3 \cdot 89 \)
\( J_4 \)  \(=\) \(1482\) \(=\)  \( 2 \cdot 3 \cdot 13 \cdot 19 \)
\( J_6 \)  \(=\) \(-2884\) \(=\)  \( - 2^{2} \cdot 7 \cdot 103 \)
\( J_8 \)  \(=\) \(-741588\) \(=\)  \( - 2^{2} \cdot 3 \cdot 29 \cdot 2131 \)
\( J_{10} \)  \(=\) \(472392\) \(=\)  \( 2^{3} \cdot 3^{10} \)
\( g_1 \)  \(=\) \(5584059449/1944\)
\( g_2 \)  \(=\) \(174127343/2916\)
\( g_3 \)  \(=\) \(-5711041/13122\)

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

magma: [C![0,-2,1],C![0,1,1],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: \(\Z/{3}\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.0.3359232.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 12.04808 \)
Tamagawa product: \( 9 \)
Torsion order:\( 9 \)
Leading coefficient: \( 1.338675 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)
\(3\) \(8\) \(10\) \(3\) \(1\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_3)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 3.1.243.1 with defining polynomial:
  \(x^{3} - 3\)

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
  \(g_4 = 297 b^{2}\)
  \(g_6 = -14337\)
   Conductor norm: 216
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 513 b^{2}\)
  \(g_6 = -34749\)
   Conductor norm: 216

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 \(K = \Q (a) \simeq \) 6.0.177147.2 with defining polynomial \(x^{6} + 3\)

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 \(-\frac{1}{2} a^{3} + \frac{1}{2}\) 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: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.243.1 with generator \(\frac{1}{2} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{3} - 3\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 3.1.243.1 with generator \(-\frac{1}{2} a^{5} + \frac{1}{2} a^{2}\) with minimal polynomial \(x^{3} - 3\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 3.1.243.1 with generator \(-a^{2}\) with minimal polynomial \(x^{3} - 3\):

\(\End (J_{F})\)\(\simeq\)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