Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| 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: | 8C0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}3&232\\28&183\end{bmatrix}$, $\begin{bmatrix}54&155\\83&6\end{bmatrix}$, $\begin{bmatrix}67&94\\36&221\end{bmatrix}$, $\begin{bmatrix}147&98\\106&83\end{bmatrix}$, $\begin{bmatrix}149&180\\26&31\end{bmatrix}$, $\begin{bmatrix}182&179\\179&70\end{bmatrix}$, $\begin{bmatrix}194&45\\105&94\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $23592960$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.48.0-16.e.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.e.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.e.2.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.e.2.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.e.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.e.1.30 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.e.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.e.2.29 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.f.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.f.1.14 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.f.2.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.f.2.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.f.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.f.1.29 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.f.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.f.2.30 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.g.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.g.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.g.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.g.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.h.1.7 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-16.h.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.h.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.h.1.23 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.i.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.k.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.1.27 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.2.28 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.1.32 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.2.17 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.2.40 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.1.28 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.2.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.2.26 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.1.24 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.2.48 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.o.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.o.1.20 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.o.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.o.1.20 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.p.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.p.1.24 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.p.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.p.1.25 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.q.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.r.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.ba.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.ba.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.ba.2.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.ba.2.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.bb.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.bb.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.bb.2.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-8.bb.2.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bh.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bj.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.bj.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bl.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.bl.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bn.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.bn.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.bp.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.by.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.by.1.16 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.by.2.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.by.2.16 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bz.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bz.1.16 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bz.2.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-24.bz.2.16 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.ca.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.ca.1.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.ca.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.ca.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.cb.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.cb.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.cb.2.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-40.cb.2.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.dd.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.df.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.dh.1.12 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.dj.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ei.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ei.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ei.2.14 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ei.2.23 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ej.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ej.1.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ej.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-120.ej.2.31 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-16.a.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-16.a.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-48.a.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-48.a.1.23 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.a.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.a.1.24 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.a.1.16 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.a.1.17 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-16.b.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-16.b.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-48.b.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-48.b.1.22 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.b.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.b.1.20 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.b.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.b.1.24 | $240$ | $2$ | $2$ | $1$ |
| 240.72.2-24.cj.1.12 | $240$ | $3$ | $3$ | $2$ |
| 240.96.1-24.ir.1.8 | $240$ | $4$ | $4$ | $1$ |
| 240.120.4-40.bl.1.6 | $240$ | $5$ | $5$ | $4$ |
| 240.144.3-40.bx.1.7 | $240$ | $6$ | $6$ | $3$ |
| 240.240.7-40.cj.1.12 | $240$ | $10$ | $10$ | $7$ |