Properties

Label 576.a.576.1
Conductor 576
Discriminant -576
Mordell-Weil group \(\Z/{10}\Z\)
Sato-Tate group $E_2$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 3x^4 + 3x^2 - 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, -1], R![1, 1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, -1]), R([1, 1, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -2, 3, 0, 3, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(576\) = \( 2^{6} \cdot 3^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-576\) = \( - 2^{6} \cdot 3^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-544\) =  \( - 2^{5} \cdot 17 \)
\( I_4 \)  = \(7936\) =  \( 2^{8} \cdot 31 \)
\( I_6 \)  = \(-1339392\) =  \( - 2^{12} \cdot 3 \cdot 109 \)
\( I_{10} \)  = \(-2359296\) =  \( - 2^{18} \cdot 3^{2} \)
\( J_2 \)  = \(-68\) =  \( - 2^{2} \cdot 17 \)
\( J_4 \)  = \(110\) =  \( 2 \cdot 5 \cdot 11 \)
\( J_6 \)  = \(36\) =  \( 2^{2} \cdot 3^{2} \)
\( J_8 \)  = \(-3637\) =  \( - 3637 \)
\( J_{10} \)  = \(-576\) =  \( - 2^{6} \cdot 3^{2} \)
\( g_1 \)  = \(22717712/9\)
\( g_2 \)  = \(540430/9\)
\( g_3 \)  = \(-289\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_4$
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)

magma: [C![0,-1,1],C![0,0,1],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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0\) \(10\)

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: \( 22.39625 \)
Tamagawa product: \( 1 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.223962 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(6\) \(1\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(1 + T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_2$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 2.2.8.1-9.1-a3

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

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