Properties

Label 880.a.225280.1
Conductor 880
Discriminant -225280
Mordell-Weil group \(\Z/{2}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 + 13x^4 + 55x^3 + 76x^2 - 44$ (homogenize, simplify)
$y^2 = x^5z + 13x^4z^2 + 55x^3z^3 + 76x^2z^4 - 44z^6$ (dehomogenize, simplify)
$y^2 = x^5 + 13x^4 + 55x^3 + 76x^2 - 44$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-44, 0, 76, 55, 13, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-44, 0, 76, 55, 13, 1]), R([]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-44, 0, 76, 55, 13, 1]))
 

Invariants

Conductor: \( N \)  =  \(880\) = \( 2^{4} \cdot 5 \cdot 11 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(880,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-225280\) = \( - 2^{12} \cdot 5 \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(37472\) =  \( 2^{5} \cdot 1171 \)
\( I_4 \)  = \(28659712\) =  \( 2^{12} \cdot 6997 \)
\( I_6 \)  = \(301361299456\) =  \( 2^{15} \cdot 7 \cdot 19 \cdot 69149 \)
\( I_{10} \)  = \(-922746880\) =  \( - 2^{24} \cdot 5 \cdot 11 \)
\( J_2 \)  = \(4684\) =  \( 2^{2} \cdot 1171 \)
\( J_4 \)  = \(615622\) =  \( 2 \cdot 7 \cdot 43973 \)
\( J_6 \)  = \(103120196\) =  \( 2^{2} \cdot 83 \cdot 263 \cdot 1181 \)
\( J_8 \)  = \(26006137795\) =  \( 5 \cdot 11 \cdot 541 \cdot 874009 \)
\( J_{10} \)  = \(-225280\) =  \( - 2^{12} \cdot 5 \cdot 11 \)
\( g_1 \)  = \(-2201833501574851/220\)
\( g_2 \)  = \(-494259267301121/1760\)
\( g_3 \)  = \(-35350660170809/3520\)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0)\)

magma: [C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.2.242000.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 1.515081 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.378770 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(12\) \(4\) \(1\) \(1\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 5 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 11 T^{2} )\)

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 20.a1
  Elliptic curve 44.a1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(3\) 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 \(\Q\).