Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $8^{4}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16C5 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}47&140\\112&179\end{bmatrix}$, $\begin{bmatrix}49&0\\50&71\end{bmatrix}$, $\begin{bmatrix}59&104\\44&231\end{bmatrix}$, $\begin{bmatrix}165&136\\214&231\end{bmatrix}$, $\begin{bmatrix}167&12\\192&131\end{bmatrix}$, $\begin{bmatrix}205&18\\44&215\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.ch.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $6144$ |
| Full 240-torsion field degree: | $2949120$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 y^{2} + 2 y z - t^{2} $ |
| $=$ | $2 y z - 2 z^{2} + w^{2}$ | |
| $=$ | $12 x^{2} - w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} y^{2} - y^{4} z^{2} + 3 y^{2} z^{4} - 2 z^{6} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1/2:-1/2:1:0)$, $(0:-1/2:1/2:1:0)$, $(0:-1/2:-1/2:0:1)$, $(0:1/2:1/2:0:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 48.48.3.d.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y+z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-Y^{3}Z+YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.ch.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}Y^{2}-Y^{4}Z^{2}+3Y^{2}Z^{4}-2Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.96.1-8.k.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ |
| 240.96.1-8.k.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ |
| 240.96.3-48.c.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.c.2.23 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.d.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.d.2.23 | $240$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.384.9-48.cc.2.15 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.ci.2.15 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.cs.2.14 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.cu.2.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.ga.4.11 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gg.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gi.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gj.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gt.2.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gv.2.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.hf.2.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.hl.2.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.li.2.23 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.lo.2.31 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.qo.2.12 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.qq.1.16 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjj.2.6 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjl.2.15 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjp.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjr.2.14 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bkr.2.30 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bkt.2.32 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bnp.2.31 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bnv.2.31 | $240$ | $2$ | $2$ | $9$ |