Properties

Label 600.a.18000.1
Conductor 600
Discriminant 18000
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{6}\Z\)
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

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = 10x^5 - 18x^4 + 8x^3 + x^2 - x$ (homogenize, simplify)
$y^2 + xz^2y = 10x^5z - 18x^4z^2 + 8x^3z^3 + x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 40x^5 - 72x^4 + 32x^3 + 5x^2 - 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 8, -18, 10]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 8, -18, 10], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, -4, 5, 32, -72, 40]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -2 : 2),\, (4 : -50 : 5)\)

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]];
 

Number of rational Weierstrass points: \(4\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 18.93431 \)
Tamagawa product: \( 8 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.262976 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 20.a3
  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 \(3\) 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 \(\Q\).