Properties

Label 864.a.1728.1
Conductor $864$
Discriminant $-1728$
Mordell-Weil group \(\Z/{12}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^2$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 + x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 7x^4 + 4x^3 + 7x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 0, 1]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 0, 1], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, 7, 4, 7, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(864\) \(=\) \( 2^{5} \cdot 3^{3} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-1728\) \(=\) \( - 2^{6} \cdot 3^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(96\) \(=\)  \( 2^{5} \cdot 3 \)
\( I_4 \)  \(=\) \(180\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5 \)
\( I_6 \)  \(=\) \(5256\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 73 \)
\( I_{10} \)  \(=\) \(216\) \(=\)  \( 2^{3} \cdot 3^{3} \)
\( J_2 \)  \(=\) \(96\) \(=\)  \( 2^{5} \cdot 3 \)
\( J_4 \)  \(=\) \(264\) \(=\)  \( 2^{3} \cdot 3 \cdot 11 \)
\( J_6 \)  \(=\) \(576\) \(=\)  \( 2^{6} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-3600\) \(=\)  \( - 2^{4} \cdot 3^{2} \cdot 5^{2} \)
\( J_{10} \)  \(=\) \(1728\) \(=\)  \( 2^{6} \cdot 3^{3} \)
\( g_1 \)  \(=\) \(4718592\)
\( g_2 \)  \(=\) \(135168\)
\( g_3 \)  \(=\) \(3072\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - x^2z\) \(0\) \(12\)

2-torsion field: 4.0.432.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 18.14296 \)
Tamagawa product: \( 3 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.377978 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(3\) \(1\)
\(3\) \(3\) \(3\) \(1\) \(1 + T\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 36.a4
  Elliptic curve 24.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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(4\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)