Properties

Label 990.a.240570.1
Conductor 990
Discriminant 240570
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = 3x^5 + 28x^4 + 72x^3 + 28x^2 + 3x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 3x^5z + 28x^4z^2 + 72x^3z^3 + 28x^2z^4 + 3xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 + 113x^4 + 290x^3 + 113x^2 + 12x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 28, 72, 28, 3], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 28, 72, 28, 3]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([0, 12, 113, 290, 113, 12]))
 

Invariants

Conductor: \( N \)  =  \(990\) = \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(240570\) = \( 2 \cdot 3^{7} \cdot 5 \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(306056\) =  \( 2^{3} \cdot 67 \cdot 571 \)
\( I_4 \)  = \(27393028\) =  \( 2^{2} \cdot 13 \cdot 263 \cdot 2003 \)
\( I_6 \)  = \(2746933168904\) =  \( 2^{3} \cdot 31 \cdot 11076343423 \)
\( I_{10} \)  = \(985374720\) =  \( 2^{13} \cdot 3^{7} \cdot 5 \cdot 11 \)
\( J_2 \)  = \(38257\) =  \( 67 \cdot 571 \)
\( J_4 \)  = \(60697908\) =  \( 2^{2} \cdot 3^{2} \cdot 37 \cdot 45569 \)
\( J_6 \)  = \(127876480380\) =  \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 11 \cdot 2392003 \)
\( J_8 \)  = \(301983618580299\) =  \( 3^{4} \cdot 19^{2} \cdot 20117 \cdot 513367 \)
\( J_{10} \)  = \(240570\) =  \( 2 \cdot 3^{7} \cdot 5 \cdot 11 \)
\( g_1 \)  = \(81951056110393451083057/240570\)
\( g_2 \)  = \(188813894774599018858/13365\)
\( g_3 \)  = \(7001861848004294/9\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
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: \((1 : 0 : 0),\, (0 : 0 : 1)\)

magma: [C![0,0,1],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: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + 19xz + 4z^2\) \(=\) \(0,\) \(8y\) \(=\) \(15xz^2 + 4z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 + 14xz + 3z^2\) \(=\) \(0,\) \(6y\) \(=\) \(11xz^2 + 3z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{10}, \sqrt{33})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 3.087468 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.385933 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(7\) \(2\) \(2\) \(( 1 + T )^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 2 T + 5 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 11 T^{2} )\)

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 15.a2
  Elliptic curve 66.b2

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 \(\Q\).