Properties

Label 1152.a.147456.1
Conductor 1152
Discriminant 147456
Mordell-Weil group \(\Z/{8}\Z\)
Sato-Tate group $J(E_1)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^6 - 2x^4 + 2x^2 - 1$ (homogenize, simplify)
$y^2 = x^6 - 2x^4z^2 + 2x^2z^4 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 2x^2 - 1$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  =  \(1152\) = \( 2^{7} \cdot 3^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1152,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(147456\) = \( 2^{14} \cdot 3^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(4864\) =  \( 2^{8} \cdot 19 \)
\( I_4 \)  = \(111616\) =  \( 2^{10} \cdot 109 \)
\( I_6 \)  = \(179208192\) =  \( 2^{15} \cdot 3 \cdot 1823 \)
\( I_{10} \)  = \(603979776\) =  \( 2^{26} \cdot 3^{2} \)
\( J_2 \)  = \(608\) =  \( 2^{5} \cdot 19 \)
\( J_4 \)  = \(14240\) =  \( 2^{5} \cdot 5 \cdot 89 \)
\( J_6 \)  = \(405504\) =  \( 2^{12} \cdot 3^{2} \cdot 11 \)
\( J_8 \)  = \(10942208\) =  \( 2^{8} \cdot 42743 \)
\( J_{10} \)  = \(147456\) =  \( 2^{14} \cdot 3^{2} \)
\( g_1 \)  = \(5071050752/9\)
\( g_2 \)  = \(195344320/9\)
\( g_3 \)  = \(1016576\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)

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

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: \(\Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(8\)

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: \( 7.270694 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.454418 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(14\) \(7\) \(4\) \(1\)
\(3\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_1)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 24.a5
  Elliptic curve 48.a5

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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 \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)