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This is a model for the modular curve $X_1(18)$. The integer $18$ is the largest $N \in \mathbb{N}$ such that $X_1(N)$ has genus $2$.

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 2, 2, 1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 2, 2, 1]), R([1, 1, 0, 1]))

$y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$324$$ = $$2^{2} \cdot 3^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-648$$ = $$-1 \cdot 2^{3} \cdot 3^{4}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-120$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 5$$ $$I_4$$ = $$3780$$ = $$2^{2} \cdot 3^{3} \cdot 5 \cdot 7$$ $$I_6$$ = $$-18360$$ = $$-1 \cdot 2^{3} \cdot 3^{3} \cdot 5 \cdot 17$$ $$I_{10}$$ = $$-2654208$$ = $$-1 \cdot 2^{15} \cdot 3^{4}$$ $$J_2$$ = $$-15$$ = $$-1 \cdot 3 \cdot 5$$ $$J_4$$ = $$-30$$ = $$-1 \cdot 2 \cdot 3 \cdot 5$$ $$J_6$$ = $$-140$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 7$$ $$J_8$$ = $$300$$ = $$2^{2} \cdot 3 \cdot 5^{2}$$ $$J_{10}$$ = $$-648$$ = $$-1 \cdot 2^{3} \cdot 3^{4}$$ $$g_1$$ = $$9375/8$$ $$g_2$$ = $$-625/4$$ $$g_3$$ = $$875/18$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_6$$ (GAP id : [6,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_6$$ (GAP id : [12,4])

## 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![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

All rational points: (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 25.521768502968100246680260022 Tamagawa numbers: 3 (p = 2), 1 (p = 3) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{21}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial:
$$x^{3} - 3 x - 1$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 3.3.81.1-8.1-a1

### Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$