Properties

Label 450.a.36450.1
Conductor $450$
Discriminant $36450$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{12}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more about

Show commands for: SageMath / Magma

This curve is isomorphic to the quotient of the modular curve $X_0(30)$ by the involution $W_{10}$; see [MR:1373390].

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 - 4x^4 - 9x^3 + 28x^2 - 6x - 16$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 4x^4z^2 - 9x^3z^3 + 28x^2z^4 - 6xz^5 - 16z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 16x^4 - 34x^3 + 112x^2 - 24x - 63$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-16, -6, 28, -9, -4, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-16, -6, 28, -9, -4, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-63, -24, 112, -34, -16, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(450\) \(=\) \( 2 \cdot 3^{2} \cdot 5^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(36450\) \(=\) \( 2 \cdot 3^{6} \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(23444\) \(=\)  \( 2^{2} \cdot 5861 \)
\( I_4 \)  \(=\) \(212089\) \(=\)  \( 131 \cdot 1619 \)
\( I_6 \)  \(=\) \(1627179821\) \(=\)  \( 29 \cdot 37 \cdot 59 \cdot 25703 \)
\( I_{10} \)  \(=\) \(4665600\) \(=\)  \( 2^{8} \cdot 3^{6} \cdot 5^{2} \)
\( J_2 \)  \(=\) \(5861\) \(=\)  \( 5861 \)
\( J_4 \)  \(=\) \(1422468\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 13171 \)
\( J_6 \)  \(=\) \(457836300\) \(=\)  \( 2^{2} \cdot 3^{5} \cdot 5^{2} \cdot 83 \cdot 227 \)
\( J_8 \)  \(=\) \(164990835819\) \(=\)  \( 3^{5} \cdot 19 \cdot 71 \cdot 503317 \)
\( J_{10} \)  \(=\) \(36450\) \(=\)  \( 2 \cdot 3^{6} \cdot 5^{2} \)
\( g_1 \)  \(=\) \(6916057684302385301/36450\)
\( g_2 \)  \(=\) \(5303516319500302/675\)
\( g_3 \)  \(=\) \(1294426477922/3\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)\)

magma: [C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z/{2}\Z \times \Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - 2xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2 + 2z^3\) \(0\) \(12\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 18.77899 \)
Tamagawa product: \( 6 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.195614 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(3\) \(2\) \(6\) \(6\) \(( 1 - T )( 1 + T )\)
\(5\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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 15.a6
  Elliptic curve 30.a6

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 \(\Q\).