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: | $\begin{bmatrix}30&197\\229&206\end{bmatrix}$, $\begin{bmatrix}89&102\\62&97\end{bmatrix}$, $\begin{bmatrix}144&221\\191&90\end{bmatrix}$, $\begin{bmatrix}155&142\\218&167\end{bmatrix}$, $\begin{bmatrix}159&50\\26&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bl.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 w + z^{2} + z w + w^{2} $ |
| $=$ | $12 x^{2} + y^{2} + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 15 x^{2} z^{2} + 11 x z^{3} + 3 y^{2} z^{2} + 7 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3}\cdot\frac{18432xy^{10}w+472320xy^{8}w^{3}+4078080xy^{6}w^{5}+12581568xy^{4}w^{7}+7879032xy^{2}w^{9}+2985255xw^{11}-1024y^{12}-25344y^{10}w^{2}-213696y^{8}w^{4}-616896y^{6}w^{6}+28188y^{4}w^{8}-995085y^{2}w^{10}-746496w^{12}}{w^{2}y^{2}(480xy^{6}w-1152xy^{4}w^{3}+594xy^{2}w^{5}-81xw^{7}-32y^{8}+216y^{6}w^{2}-162y^{4}w^{4}+27y^{2}w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+8X^{3}Z+15X^{2}Z^{2}+3Y^{2}Z^{2}+11XZ^{3}+7Z^{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.bz.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.e.1.3 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.e.1.21 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-24.bz.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-16.b.1.14 | $240$ | $2$ | $2$ | $1$ | $?$ | 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.l.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bc.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bk.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cc.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cq.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.cr.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dd.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dk.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.hy.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ic.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.io.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.is.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jo.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jw.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ku.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.lc.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.288.9-48.fb.2.10 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.bbc.1.29 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.cv.1.13 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |