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: | 16G1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}14&125\\75&164\end{bmatrix}$, $\begin{bmatrix}80&129\\47&134\end{bmatrix}$, $\begin{bmatrix}107&108\\106&1\end{bmatrix}$, $\begin{bmatrix}121&0\\116&157\end{bmatrix}$, $\begin{bmatrix}128&133\\39&106\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bm.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 $ | $=$ | $ 12 x y + 3 y^{2} - w^{2} $ |
| $=$ | $12 x^{2} - 3 x y - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 3 x^{2} y^{2} - 9 x^{2} z^{2} + 9 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{193536y^{2}z^{10}+6912y^{2}z^{8}w^{2}-2726784y^{2}z^{6}w^{4}-12053664y^{2}z^{4}w^{6}-9439308y^{2}z^{2}w^{8}-786429y^{2}w^{10}-131072z^{12}-196608z^{10}w^{2}+303360z^{8}w^{4}+191744z^{6}w^{6}-173568z^{4}w^{8}+1049280z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(192y^{2}z^{6}+528y^{2}z^{4}w^{2}+84y^{2}z^{2}w^{4}+3y^{2}w^{6}-512z^{6}w^{2}-272z^{4}w^{4}-32z^{2}w^{6}-w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bm.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{2}Y^{2}-9X^{2}Z^{2}+9Z^{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.by.2.15 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.f.2.1 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.f.2.21 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-24.by.2.5 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-16.b.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | 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.j.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bd.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bo.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cb.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cn.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cu.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dg.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dh.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.hz.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.id.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ip.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.it.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jp.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jx.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kv.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ld.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.288.9-48.fk.2.12 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.bbd.1.25 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.da.2.9 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |