Properties

Label 882.a.63504.1
Conductor $882$
Discriminant $63504$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 + x^4 + x^3 + 3x^2 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + x^4z^2 + x^3z^3 + 3x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 5x^4 + 6x^3 + 13x^2 + 12x + 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 3, 1, 1, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 3, 1, 1, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 12, 13, 6, 5, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(882\) \(=\) \( 2 \cdot 3^{2} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(63504\) \(=\) \( 2^{4} \cdot 3^{4} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(548\) \(=\)  \( 2^{2} \cdot 137 \)
\( I_4 \)  \(=\) \(6049\) \(=\)  \( 23 \cdot 263 \)
\( I_6 \)  \(=\) \(662961\) \(=\)  \( 3 \cdot 13 \cdot 89 \cdot 191 \)
\( I_{10} \)  \(=\) \(8128512\) \(=\)  \( 2^{11} \cdot 3^{4} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(137\) \(=\)  \( 137 \)
\( J_4 \)  \(=\) \(530\) \(=\)  \( 2 \cdot 5 \cdot 53 \)
\( J_6 \)  \(=\) \(6336\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 11 \)
\( J_8 \)  \(=\) \(146783\) \(=\)  \( 7 \cdot 13 \cdot 1613 \)
\( J_{10} \)  \(=\) \(63504\) \(=\)  \( 2^{4} \cdot 3^{4} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(48261724457/63504\)
\( g_2 \)  \(=\) \(681408545/31752\)
\( g_3 \)  \(=\) \(825836/441\)

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

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

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: \(\Z/{2}\Z \times \Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + 5xz + 2z^2\) \(=\) \(0,\) \(8y\) \(=\) \(xz^2 + 2z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(8xz^2 + 7z^3\) \(0\) \(8\)

2-torsion field: \(\Q(\sqrt{-7}) \)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 12.54262 \)
Tamagawa product: \( 8 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.391956 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(4\) \(4\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(2\) \(4\) \(2\) \(( 1 - T )( 1 + T )\)
\(7\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 42.a5
  Elliptic curve 21.a6

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).