Properties

Label 600.a.18000.1
Conductor 600
Discriminant 18000
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

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 8, -18, 10], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 8, -18, 10]), R([0, 1]))
 

$y^2 + xy = 10x^5 - 18x^4 + 8x^3 + x^2 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 600 \)  =  \( 2^{3} \cdot 3 \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(18000\)  =  \( 2^{4} \cdot 3^{2} \cdot 5^{3} \)

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);
 

\( I_2 \)  =  \(5504\)  =  \( 2^{7} \cdot 43 \)
\( I_4 \)  =  \(381184\)  =  \( 2^{8} \cdot 1489 \)
\( I_6 \)  =  \(730242816\)  =  \( 2^{8} \cdot 3 \cdot 950837 \)
\( I_{10} \)  =  \(73728000\)  =  \( 2^{16} \cdot 3^{2} \cdot 5^{3} \)
\( J_2 \)  =  \(688\)  =  \( 2^{4} \cdot 43 \)
\( J_4 \)  =  \(15752\)  =  \( 2^{3} \cdot 11 \cdot 179 \)
\( J_6 \)  =  \(244900\)  =  \( 2^{2} \cdot 5^{2} \cdot 31 \cdot 79 \)
\( J_8 \)  =  \(-19908576\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \cdot 69127 \)
\( J_{10} \)  =  \(18000\)  =  \( 2^{4} \cdot 3^{2} \cdot 5^{3} \)
\( g_1 \)  =  \(9634345320448/1125\)
\( g_2 \)  =  \(320612931584/1125\)
\( g_3 \)  =  \(289804864/45\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

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![0,0,1],C![1,-2,2],C![1,-1,1],C![1,0,0],C![1,0,1],C![4,-50,5]];
 

All rational points: (0 : 0 : 1), (1 : -2 : 2), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1), (4 : -50 : 5)

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

Number of rational Weierstrass points: \(4\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 18.934318681069611836111049526

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

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

Torsion: \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{6}\Z\)

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

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 20.a3
  Elliptic curve 30.a6

Endomorphisms

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 \(\Q\).