# 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

# Related objects

Show commands for: SageMath / Magma

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.

## 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)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, 0, 1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, 0, 1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([1, -2, 1, 2, 2, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

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

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

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

## 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 Cluster picture
$$67$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## 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 $$\Q$$.