Properties

Label 262144.c.524288.1
Conductor $262144$
Discriminant $524288$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 2x^3 + 2x$ (homogenize, simplify)
$y^2 = x^5z - 2x^3z^3 + 2xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - 2x^3 + 2x$ (homogenize, minimize)

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

Invariants

Conductor: \( N \)  \(=\)  \(262144\) \(=\) \( 2^{18} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(262144,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(524288\) \(=\) \( 2^{19} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(26\) \(=\)  \( 2 \cdot 13 \)
\( I_4 \)  \(=\) \(-2\) \(=\)  \( -2 \)
\( I_6 \)  \(=\) \(40\) \(=\)  \( 2^{3} \cdot 5 \)
\( I_{10} \)  \(=\) \(2\) \(=\)  \( 2 \)
\( J_2 \)  \(=\) \(208\) \(=\)  \( 2^{4} \cdot 13 \)
\( J_4 \)  \(=\) \(1888\) \(=\)  \( 2^{5} \cdot 59 \)
\( J_6 \)  \(=\) \(-2304\) \(=\)  \( - 2^{8} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-1010944\) \(=\)  \( - 2^{8} \cdot 11 \cdot 359 \)
\( J_{10} \)  \(=\) \(524288\) \(=\)  \( 2^{19} \)
\( g_1 \)  \(=\) \(742586\)
\( g_2 \)  \(=\) \(129623/4\)
\( g_3 \)  \(=\) \(-1521/8\)

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1/2 : 1),\, (1 : 1/2 : 1)\)

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1]]; // minimal model
 
magma: [C![0,0,1],C![1,-1/2,1],C![1,0,0],C![1,1/2,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.759196\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.759196\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1/2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-1/2z^3\) \(0.759196\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 4.0.512.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.759196 \)
Real period: \( 7.048010 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.675411 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(18\) \(19\) \(2\) \(1\)

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?
\(2\) 2.90.3 yes
\(3\) 3.270.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.2.2048.1 with defining polynomial:
  \(x^{4} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 96\)
  \(g_6 = 1792 b^{3} + 2304 b\)
   Conductor norm: 16384
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 96\)
  \(g_6 = -1792 b^{3} - 2304 b\)
   Conductor norm: 16384

magma: HeuristicDecompositionFactors(C);
 

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 \(K = \Q (a) \simeq \) 8.0.16777216.2 with defining polynomial \(x^{8} - 4 x^{6} + 8 x^{4} - 4 x^{2} + 1\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(4\) 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 \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{2}{3} a^{6} - \frac{7}{3} a^{4} + \frac{14}{3} a^{2} - \frac{4}{3}\) with minimal polynomial \(x^{2} + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: E_4
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(-\frac{1}{3} a^{6} + \frac{2}{3} a^{4} - \frac{1}{3} a^{2} - \frac{4}{3}\) 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_2)
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(a^{6} - 4 a^{4} + 7 a^{2} - 2\) 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_2)
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{4}{3} a^{7} - \frac{14}{3} a^{5} + \frac{25}{3} a^{3} - \frac{5}{3} a\) with minimal polynomial \(x^{4} - 2\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 4.0.2048.1 with generator \(-a^{7} + 4 a^{5} - 8 a^{3} + 3 a\) with minimal polynomial \(x^{4} + 2\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.2048.1 with generator \(\frac{2}{3} a^{7} - \frac{7}{3} a^{5} + \frac{11}{3} a^{3} + \frac{2}{3} a\) with minimal polynomial \(x^{4} + 2\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{1}{3} a^{7} - \frac{5}{3} a^{5} + \frac{10}{3} a^{3} - \frac{8}{3} a\) with minimal polynomial \(x^{4} - 2\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) \(\Q(\zeta_{8})\) with generator \(\frac{1}{3} a^{6} - \frac{5}{3} a^{4} + \frac{10}{3} a^{2} - \frac{5}{3}\) with minimal polynomial \(x^{4} + 1\):

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