Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}1&120\\135&89\end{bmatrix}$, $\begin{bmatrix}59&176\\119&237\end{bmatrix}$, $\begin{bmatrix}97&232\\127&93\end{bmatrix}$, $\begin{bmatrix}121&40\\82&131\end{bmatrix}$, $\begin{bmatrix}199&32\\128&155\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.s.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $5898240$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x y + 6 x z + w^{2} $ |
| $=$ | $12 x^{2} - y^{2} - 3 y z - 3 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 108 x^{4} - x^{2} y^{2} - x y z^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3}\cdot\frac{79272xz^{9}w^{2}+196704xz^{5}w^{6}-18432xzw^{10}-19656y^{2}z^{10}-27180y^{2}z^{6}w^{4}-17856y^{2}z^{2}w^{8}-58968yz^{11}-29196yz^{7}w^{4}-38784yz^{3}w^{8}-39285z^{12}+49572z^{8}w^{4}+6432z^{4}w^{8}-512w^{12}}{w^{4}(108xz^{5}w^{2}+60xzw^{6}+9y^{2}z^{6}+27y^{2}z^{2}w^{4}+27yz^{7}+57yz^{3}w^{4}+27z^{8}+69z^{4}w^{4}+4w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.s.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 108X^{4}-X^{2}Y^{2}-XYZ^{2}-Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.48.1-16.b.1.15 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.0-24.bh.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.g.1.8 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.g.1.29 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-24.bh.1.8 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-16.b.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 240.192.1-48.cn.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cn.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.co.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.co.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cp.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cp.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cq.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cq.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gr.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gr.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gs.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gs.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gt.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gt.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gu.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gu.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.288.9-48.co.1.11 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.zr.1.27 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.bi.1.21 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |