# Properties

 Label 529.a.529.1 Conductor $529$ Discriminant $529$ Mordell-Weil group $$\Z/{11}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{RM}$$ $$\End(J) \otimes \Q$$ $$\mathsf{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

The Jacobian of this curve is isogenous to that of the modular curve $X_0(23)$ (which has discriminant $23^6$).

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^5$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^5z$ (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, 0, 0, 0, 0, -1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 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: $$N$$ $$=$$ $$529$$ $$=$$ $$23^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$529$$ $$=$$ $$23^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$284$$ $$=$$ $$2^{2} \cdot 71$$ $$I_4$$ $$=$$ $$2401$$ $$=$$ $$7^{4}$$ $$I_6$$ $$=$$ $$246639$$ $$=$$ $$3 \cdot 19 \cdot 4327$$ $$I_{10}$$ $$=$$ $$-67712$$ $$=$$ $$- 2^{7} \cdot 23^{2}$$ $$J_2$$ $$=$$ $$71$$ $$=$$ $$71$$ $$J_4$$ $$=$$ $$110$$ $$=$$ $$2 \cdot 5 \cdot 11$$ $$J_6$$ $$=$$ $$-624$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 13$$ $$J_8$$ $$=$$ $$-14101$$ $$=$$ $$- 59 \cdot 239$$ $$J_{10}$$ $$=$$ $$-529$$ $$=$$ $$- 23^{2}$$ $$g_1$$ $$=$$ $$-1804229351/529$$ $$g_2$$ $$=$$ $$-39370210/529$$ $$g_3$$ $$=$$ $$3145584/529$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

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: $$\Z/{11}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^2z$$ $$0$$ $$11$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$30.06025$$ Tamagawa product: $$1$$ Torsion order: $$11$$ Leading coefficient: $$0.248431$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$23$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )^{2}$$

## 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 $$\Q$$.