Properties

Label 450.a.36450.1
Conductor 450
Discriminant 36450
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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This curve is isomorphic to the quotient of the modular curve $X_0(30)$ by the involution $W_{10}$; see [MR:1373390].

Minimal equation

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

$y^2 + (x^3 + 1)y = x^5 - 4x^4 - 9x^3 + 28x^2 - 6x - 16$

Invariants

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

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(46888\)  =  \( 2^{3} \cdot 5861 \)
\( I_4 \)  =  \(848356\)  =  \( 2^{2} \cdot 131 \cdot 1619 \)
\( I_6 \)  =  \(13017438568\)  =  \( 2^{3} \cdot 29 \cdot 37 \cdot 59 \cdot 25703 \)
\( I_{10} \)  =  \(149299200\)  =  \( 2^{13} \cdot 3^{6} \cdot 5^{2} \)
\( J_2 \)  =  \(5861\)  =  \( 5861 \)
\( J_4 \)  =  \(1422468\)  =  \( 2^{2} \cdot 3^{3} \cdot 13171 \)
\( J_6 \)  =  \(457836300\)  =  \( 2^{2} \cdot 3^{5} \cdot 5^{2} \cdot 83 \cdot 227 \)
\( J_8 \)  =  \(164990835819\)  =  \( 3^{5} \cdot 19 \cdot 71 \cdot 503317 \)
\( J_{10} \)  =  \(36450\)  =  \( 2 \cdot 3^{6} \cdot 5^{2} \)
\( g_1 \)  =  \(6916057684302385301/36450\)
\( g_2 \)  =  \(5303516319500302/675\)
\( g_3 \)  =  \(1294426477922/3\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

Rational points

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);

This curve is locally solvable everywhere.

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

All rational points: (1 : -1 : 0), (1 : 0 : 0), (2 : -5 : 1), (2 : -4 : 1)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 6 (p = 3), 1 (p = 5)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

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 \(\GL_2\)-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 \(\Q\).