Properties

Label 363.a.11979.1
Conductor 363
Discriminant -11979
Mordell-Weil group \(\Z/{2}\Z \times \Z/{10}\Z\)
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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The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the involution $W_3$ (see [MR:1373390]), which has discriminant $3\cdot 11^9$.

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z + 2x^3z^3 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 2, 0, 1], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 2, 0, 1]), R([1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 8, 18, 8, 1, 4]))
 

Invariants

Conductor: \( N \)  =  \(363\) = \( 3 \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-11979\) = \( - 3^{2} \cdot 11^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1376\) =  \( 2^{5} \cdot 43 \)
\( I_4 \)  = \(-49088\) =  \( - 2^{6} \cdot 13 \cdot 59 \)
\( I_6 \)  = \(-33691712\) =  \( - 2^{6} \cdot 19 \cdot 103 \cdot 269 \)
\( I_{10} \)  = \(-49065984\) =  \( - 2^{12} \cdot 3^{2} \cdot 11^{3} \)
\( J_2 \)  = \(172\) =  \( 2^{2} \cdot 43 \)
\( J_4 \)  = \(1744\) =  \( 2^{4} \cdot 109 \)
\( J_6 \)  = \(45841\) =  \( 45841 \)
\( J_8 \)  = \(1210779\) =  \( 3^{2} \cdot 29 \cdot 4639 \)
\( J_{10} \)  = \(-11979\) =  \( - 3^{2} \cdot 11^{3} \)
\( g_1 \)  = \(-150536645632/11979\)
\( g_2 \)  = \(-8874253312/11979\)
\( g_3 \)  = \(-1356160144/11979\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (-1 : -34 : 4)\)

magma: [C![-1,-34,4],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 5xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-15xz^2 - 6z^3\) \(0\) \(10\)

2-torsion field: 3.1.44.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 18.97059 \)
Tamagawa product: \( 4 \)
Torsion order:\( 20 \)
Leading coefficient: \( 0.189705 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 11.a3
  Elliptic curve 33.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 \(\Q\).