Properties

Label 484.a.1936.1
Conductor 484
Discriminant -1936
Mordell-Weil group \(\Z/{15}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 0, 2, 0, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 4, 0, 8, 0, 4]))
 

$y^2 + y = x^6 + 2x^4 + x^2$ (homogenize, simplify)
$y^2 + z^3y = x^6 + 2x^4z^2 + x^2z^4$ (dehomogenize, simplify)
$y^2 = 4x^6 + 8x^4 + 4x^2 + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(484\) = \( 2^{2} \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-1936\) = \( - 2^{4} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(-1472\) =  \( - 2^{6} \cdot 23 \)
\( I_4 \)  = \(2368\) =  \( 2^{6} \cdot 37 \)
\( I_6 \)  = \(-369152\) =  \( - 2^{9} \cdot 7 \cdot 103 \)
\( I_{10} \)  = \(-7929856\) =  \( - 2^{16} \cdot 11^{2} \)
\( J_2 \)  = \(-184\) =  \( - 2^{3} \cdot 23 \)
\( J_4 \)  = \(1386\) =  \( 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
\( J_6 \)  = \(-15040\) =  \( - 2^{6} \cdot 5 \cdot 47 \)
\( J_8 \)  = \(211591\) =  \( 457 \cdot 463 \)
\( J_{10} \)  = \(-1936\) =  \( - 2^{4} \cdot 11^{2} \)
\( g_1 \)  = \(13181630464/121\)
\( g_2 \)  = \(49057344/11\)
\( g_3 \)  = \(31824640/121\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$

Rational points

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

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

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{15}\Z\)

Generator Height Order
\(2x^2 - xz + z^2\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 - 3z^3\) \(0\) \(15\)

2-torsion field: 6.0.30976.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 15.31896 \)
Tamagawa product: \( 3 \)
Torsion order:\( 15 \)
Leading coefficient: \( 0.204252 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(2\) \(3\) \(1 + 2 T + 2 T^{2}\)
\(11\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 44.a2
  Elliptic curve 11.a3

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 \(\Q\).