Properties

Label 4489.a.4489.1
Conductor 4489
Discriminant 4489
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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This is a model for the quotient of the modular curve $X_0(67)$ by its Atkin-Lehner involution $w_{67}$. This is the first genus-$2$ modular curve $X_0(N)/w_N$ for $N$ prime.

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = x^5 - x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 2x^4 + 2x^3 + x^2 - 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, 0, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, 0, 1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -2, 1, 2, 2, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(4489\) = \( 67^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(4489\) = \( 67^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-568\) =  \( - 2^{3} \cdot 71 \)
\( I_4 \)  = \(5476\) =  \( 2^{2} \cdot 37^{2} \)
\( I_6 \)  = \(-1017496\) =  \( - 2^{3} \cdot 193 \cdot 659 \)
\( I_{10} \)  = \(18386944\) =  \( 2^{12} \cdot 67^{2} \)
\( J_2 \)  = \(-71\) =  \( - 71 \)
\( J_4 \)  = \(153\) =  \( 3^{2} \cdot 17 \)
\( J_6 \)  = \(-187\) =  \( - 11 \cdot 17 \)
\( J_8 \)  = \(-2533\) =  \( - 17 \cdot 149 \)
\( J_{10} \)  = \(4489\) =  \( 67^{2} \)
\( g_1 \)  = \(-1804229351/4489\)
\( g_2 \)  = \(-54760383/4489\)
\( g_3 \)  = \(-942667/4489\)

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

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 1 : 1)\) \((1 : -3 : 1)\) \((1 : -3 : 2)\) \((1 : -10 : 2)\)

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-10,2],C![1,-3,1],C![1,-3,2],C![1,-1,0],C![1,0,0],C![1,0,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.133201\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.096171\) \(\infty\)

2-torsion field: 5.1.287296.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.011438 \)
Real period: \( 20.46511 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.234099 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(67\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).