Properties

Label 363.a.43923.1
Conductor 363
Discriminant -43923
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 + x^2y = 11x^5 - 13x^4 - 7x^3 + 10x^2 + x - 2$ (homogenize, simplify)
$y^2 + x^2zy = 11x^5z - 13x^4z^2 - 7x^3z^3 + 10x^2z^4 + xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = 44x^5 - 51x^4 - 28x^3 + 40x^2 + 4x - 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 1, 10, -7, -13, 11]), R([0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 1, 10, -7, -13, 11], R![0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, 4, 40, -28, -51, 44]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(363\) = \( 3 \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-43923\) = \( - 3 \cdot 11^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(44384\) =  \( 2^{5} \cdot 19 \cdot 73 \)
\( I_4 \)  = \(409792\) =  \( 2^{6} \cdot 19 \cdot 337 \)
\( I_6 \)  = \(5649542080\) =  \( 2^{6} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \cdot 10211 \)
\( I_{10} \)  = \(-179908608\) =  \( - 2^{12} \cdot 3 \cdot 11^{4} \)
\( J_2 \)  = \(5548\) =  \( 2^{2} \cdot 19 \cdot 73 \)
\( J_4 \)  = \(1278244\) =  \( 2^{2} \cdot 11^{2} \cdot 19 \cdot 139 \)
\( J_6 \)  = \(392069161\) =  \( 11^{2} \cdot 19 \cdot 170539 \)
\( J_8 \)  = \(135322995423\) =  \( 3 \cdot 7 \cdot 11^{2} \cdot 19^{2} \cdot 29 \cdot 5087 \)
\( J_{10} \)  = \(-43923\) =  \( - 3 \cdot 11^{4} \)
\( g_1 \)  = \(-5256325630316243968/43923\)
\( g_2 \)  = \(-1804005053317888/363\)
\( g_3 \)  = \(-99735603013264/363\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
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: \((1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)\)

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

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(11x^2 - 3xz - 6z^2\) \(=\) \(0,\) \(11y\) \(=\) \(-2xz^2 - 4z^3\) \(0\) \(10\)

2-torsion field: 6.0.52272.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 3.794119 \)
Tamagawa product: \( 5 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.189705 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 33.a3
  Elliptic curve 11.a2

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 \(\Q\).