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Properties

Label 18.36.0.a.1
Level $18$
Index $36$
Genus $0$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $18$ $\SL_2$-level: $18$
Index: $36$ $\PSL_2$-index:$36$
Genus: $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $1^{3}\cdot2^{3}\cdot9\cdot18$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 18E0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 18.36.0.1

Level structure

$\GL_2(\Z/18\Z)$-generators: $\begin{bmatrix}1&13\\0&13\end{bmatrix}$, $\begin{bmatrix}5&7\\0&7\end{bmatrix}$, $\begin{bmatrix}11&12\\0&17\end{bmatrix}$
$\GL_2(\Z/18\Z)$-subgroup: $C_{18}:C_6^2$
Contains $-I$: yes
Quadratic refinements: 18.72.0-18.a.1.1, 18.72.0-18.a.1.2, 18.72.0-18.a.1.3, 18.72.0-18.a.1.4, 36.72.0-18.a.1.1, 36.72.0-18.a.1.2, 36.72.0-18.a.1.3, 36.72.0-18.a.1.4, 36.72.0-18.a.1.5, 36.72.0-18.a.1.6, 36.72.0-18.a.1.7, 36.72.0-18.a.1.8, 36.72.0-18.a.1.9, 36.72.0-18.a.1.10, 36.72.0-18.a.1.11, 36.72.0-18.a.1.12, 72.72.0-18.a.1.1, 72.72.0-18.a.1.2, 72.72.0-18.a.1.3, 72.72.0-18.a.1.4, 72.72.0-18.a.1.5, 72.72.0-18.a.1.6, 72.72.0-18.a.1.7, 72.72.0-18.a.1.8, 72.72.0-18.a.1.9, 72.72.0-18.a.1.10, 72.72.0-18.a.1.11, 72.72.0-18.a.1.12, 72.72.0-18.a.1.13, 72.72.0-18.a.1.14, 72.72.0-18.a.1.15, 72.72.0-18.a.1.16, 90.72.0-18.a.1.1, 90.72.0-18.a.1.2, 90.72.0-18.a.1.3, 90.72.0-18.a.1.4, 126.72.0-18.a.1.1, 126.72.0-18.a.1.2, 126.72.0-18.a.1.3, 126.72.0-18.a.1.4, 180.72.0-18.a.1.1, 180.72.0-18.a.1.2, 180.72.0-18.a.1.3, 180.72.0-18.a.1.4, 180.72.0-18.a.1.5, 180.72.0-18.a.1.6, 180.72.0-18.a.1.7, 180.72.0-18.a.1.8, 180.72.0-18.a.1.9, 180.72.0-18.a.1.10, 180.72.0-18.a.1.11, 180.72.0-18.a.1.12, 198.72.0-18.a.1.1, 198.72.0-18.a.1.2, 198.72.0-18.a.1.3, 198.72.0-18.a.1.4, 234.72.0-18.a.1.1, 234.72.0-18.a.1.2, 234.72.0-18.a.1.3, 234.72.0-18.a.1.4, 252.72.0-18.a.1.1, 252.72.0-18.a.1.2, 252.72.0-18.a.1.3, 252.72.0-18.a.1.4, 252.72.0-18.a.1.5, 252.72.0-18.a.1.6, 252.72.0-18.a.1.7, 252.72.0-18.a.1.8, 252.72.0-18.a.1.9, 252.72.0-18.a.1.10, 252.72.0-18.a.1.11, 252.72.0-18.a.1.12, 306.72.0-18.a.1.1, 306.72.0-18.a.1.2, 306.72.0-18.a.1.3, 306.72.0-18.a.1.4
Cyclic 18-isogeny field degree: $1$
Cyclic 18-torsion field degree: $6$
Full 18-torsion field degree: $648$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 92 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 36 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{36}(x^{3}-2y^{3})^{3}(x^{9}-6x^{6}y^{3}-12x^{3}y^{6}-8y^{9})^{3}}{y^{18}x^{45}(x-2y)(x+y)^{2}(x^{2}-xy+y^{2})^{2}(x^{2}+2xy+4y^{2})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_0(2)$ $2$ $12$ $12$ $0$ $0$
$X_0(9)$ $9$ $3$ $3$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(6)$ $6$ $3$ $3$ $0$ $0$
$X_0(9)$ $9$ $3$ $3$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
18.72.1.a.1 $18$ $2$ $2$ $1$
18.72.1.d.1 $18$ $2$ $2$ $1$
$X_{\pm1}(18)$ $18$ $3$ $3$ $2$
18.108.2.d.2 $18$ $3$ $3$ $2$
18.108.2.e.1 $18$ $3$ $3$ $2$
18.108.4.c.1 $18$ $3$ $3$ $4$
36.72.1.a.1 $36$ $2$ $2$ $1$
36.72.1.b.1 $36$ $2$ $2$ $1$
$X_0(36)$ $36$ $2$ $2$ $1$
36.72.1.e.1 $36$ $2$ $2$ $1$
36.72.1.f.1 $36$ $2$ $2$ $1$
36.72.1.g.1 $36$ $2$ $2$ $1$
36.72.3.u.1 $36$ $2$ $2$ $3$
36.72.3.v.1 $36$ $2$ $2$ $3$
36.72.3.w.1 $36$ $2$ $2$ $3$
36.72.3.x.1 $36$ $2$ $2$ $3$
54.108.2.a.1 $54$ $3$ $3$ $2$
$X_0(54)$ $54$ $3$ $3$ $4$
54.108.6.a.1 $54$ $3$ $3$ $6$
72.72.1.a.1 $72$ $2$ $2$ $1$
72.72.1.b.1 $72$ $2$ $2$ $1$
72.72.1.c.1 $72$ $2$ $2$ $1$
72.72.1.d.1 $72$ $2$ $2$ $1$
72.72.1.g.1 $72$ $2$ $2$ $1$
72.72.1.h.1 $72$ $2$ $2$ $1$
72.72.1.i.1 $72$ $2$ $2$ $1$
72.72.1.j.1 $72$ $2$ $2$ $1$
72.72.3.cm.1 $72$ $2$ $2$ $3$
72.72.3.cn.1 $72$ $2$ $2$ $3$
72.72.3.co.1 $72$ $2$ $2$ $3$
72.72.3.cp.1 $72$ $2$ $2$ $3$
90.72.1.d.1 $90$ $2$ $2$ $1$
90.72.1.e.1 $90$ $2$ $2$ $1$
90.180.12.a.1 $90$ $5$ $5$ $12$
$X_0(90)$ $90$ $6$ $6$ $11$
90.360.23.a.1 $90$ $10$ $10$ $23$
126.72.1.i.1 $126$ $2$ $2$ $1$
126.72.1.j.1 $126$ $2$ $2$ $1$
126.108.2.d.1 $126$ $3$ $3$ $2$
126.108.2.d.2 $126$ $3$ $3$ $2$
126.108.2.e.1 $126$ $3$ $3$ $2$
126.108.2.e.2 $126$ $3$ $3$ $2$
126.108.2.f.1 $126$ $3$ $3$ $2$
126.108.2.f.2 $126$ $3$ $3$ $2$
$X_0(126)$ $126$ $8$ $8$ $17$
180.72.1.d.1 $180$ $2$ $2$ $1$
180.72.1.e.1 $180$ $2$ $2$ $1$
180.72.1.f.1 $180$ $2$ $2$ $1$
180.72.1.g.1 $180$ $2$ $2$ $1$
180.72.1.h.1 $180$ $2$ $2$ $1$
180.72.1.i.1 $180$ $2$ $2$ $1$
180.72.3.be.1 $180$ $2$ $2$ $3$
180.72.3.bf.1 $180$ $2$ $2$ $3$
180.72.3.bg.1 $180$ $2$ $2$ $3$
180.72.3.bh.1 $180$ $2$ $2$ $3$
198.72.1.b.1 $198$ $2$ $2$ $1$
198.72.1.c.1 $198$ $2$ $2$ $1$
234.72.1.i.1 $234$ $2$ $2$ $1$
234.72.1.j.1 $234$ $2$ $2$ $1$
234.108.2.d.1 $234$ $3$ $3$ $2$
234.108.2.d.2 $234$ $3$ $3$ $2$
234.108.2.e.1 $234$ $3$ $3$ $2$
234.108.2.e.2 $234$ $3$ $3$ $2$
234.108.2.f.1 $234$ $3$ $3$ $2$
234.108.2.f.2 $234$ $3$ $3$ $2$
252.72.1.f.1 $252$ $2$ $2$ $1$
252.72.1.g.1 $252$ $2$ $2$ $1$
252.72.1.h.1 $252$ $2$ $2$ $1$
252.72.1.i.1 $252$ $2$ $2$ $1$
252.72.1.j.1 $252$ $2$ $2$ $1$
252.72.1.k.1 $252$ $2$ $2$ $1$
252.72.3.cc.1 $252$ $2$ $2$ $3$
252.72.3.cd.1 $252$ $2$ $2$ $3$
252.72.3.ce.1 $252$ $2$ $2$ $3$
252.72.3.cf.1 $252$ $2$ $2$ $3$
306.72.1.b.1 $306$ $2$ $2$ $1$
306.72.1.c.1 $306$ $2$ $2$ $1$