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: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16D5 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}53&26\\200&103\end{bmatrix}$, $\begin{bmatrix}73&28\\20&87\end{bmatrix}$, $\begin{bmatrix}75&218\\196&159\end{bmatrix}$, $\begin{bmatrix}137&62\\184&231\end{bmatrix}$, $\begin{bmatrix}187&116\\168&23\end{bmatrix}$, $\begin{bmatrix}197&148\\216&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.ci.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 $ | $=$ | $ - y t + z w $ |
| $=$ | $6 x^{2} - z w$ | |
| $=$ | $8 y^{2} - 2 z^{2} - 2 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} y^{2} - x^{4} z^{2} - 36 y^{4} z^{2} + 8 y^{2} z^{4} $ |
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 |
|---|
| $(0:1/2:0:1:0)$, $(0:-1/2:0:1:0)$, $(0:-1/2:1:0:0)$, $(0:1/2:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{256z^{12}-384z^{10}t^{2}+912z^{8}t^{4}-752z^{6}t^{6}+858z^{4}t^{8}-339z^{2}t^{10}+256w^{12}-384w^{10}t^{2}+912w^{8}t^{4}-752w^{6}t^{6}+858w^{4}t^{8}-339w^{2}t^{10}+256t^{12}}{t^{4}(16z^{8}-16z^{6}t^{2}+2z^{4}t^{4}+z^{2}t^{6}+16w^{8}-16w^{6}t^{2}+2w^{4}t^{4}+w^{2}t^{6})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.ci.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}Y^{2}-X^{4}Z^{2}-36Y^{4}Z^{2}+8Y^{2}Z^{4} $ |
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.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ |
| 240.96.3-48.c.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.c.1.24 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.e.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.96.3-48.e.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.ce.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.cg.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.cq.2.4 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.cw.2.4 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.ga.3.11 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gf.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gh.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gj.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gr.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.gx.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.hh.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-48.hj.2.8 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.lk.2.6 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.lm.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.qm.2.6 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.qs.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjk.2.6 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjm.2.15 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjq.2.7 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bjs.2.14 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bkp.2.24 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bkv.2.24 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bnr.2.12 | $240$ | $2$ | $2$ | $9$ |
| 240.384.9-240.bnt.2.14 | $240$ | $2$ | $2$ | $9$ |