Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 8G0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}19&80\\124&139\end{bmatrix}$, $\begin{bmatrix}37&72\\78&49\end{bmatrix}$, $\begin{bmatrix}47&128\\180&7\end{bmatrix}$, $\begin{bmatrix}51&112\\218&69\end{bmatrix}$, $\begin{bmatrix}83&208\\220&69\end{bmatrix}$, $\begin{bmatrix}119&232\\84&89\end{bmatrix}$, $\begin{bmatrix}177&64\\16&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.i.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $11796480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 240.24.0-8.o.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.24.0-8.o.1.3 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.96.0-16.d.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-16.d.2.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-48.d.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-48.d.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.f.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.f.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.f.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.f.2.37 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.j.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.j.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.k.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.k.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.l.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-8.l.2.7 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.ba.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.ba.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.bb.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.bb.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bb.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bb.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.bc.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-24.bc.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bc.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bc.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bd.1.14 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bd.2.16 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.cy.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.cy.2.12 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.cz.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.cz.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.da.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.da.2.16 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-16.a.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-16.a.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-16.a.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.a.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.a.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.a.2.18 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.a.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.a.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.a.2.20 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.1.21 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.2.35 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-16.b.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-16.b.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-16.b.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.b.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.b.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.b.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.b.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.b.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.b.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.1.21 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.2.35 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-8.h.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-8.p.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-24.bu.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-40.bu.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-24.bv.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-40.bv.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-120.fq.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-120.fr.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-16.d.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-16.d.1.10 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-16.d.2.3 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-16.d.2.10 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-48.d.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-48.d.1.6 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-48.d.2.15 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-48.d.2.18 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-80.f.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-80.f.1.10 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-80.f.2.9 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-80.f.2.12 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.1.20 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.2.27 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.2.36 | $240$ | $2$ | $2$ | $2$ |
| 240.144.4-24.ch.1.27 | $240$ | $3$ | $3$ | $4$ |
| 240.192.3-24.cl.1.8 | $240$ | $4$ | $4$ | $3$ |
| 240.240.8-40.v.1.5 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-40.br.1.25 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-40.ch.1.8 | $240$ | $10$ | $10$ | $15$ |