Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}21&152\\145&111\end{bmatrix}$, $\begin{bmatrix}47&208\\67&157\end{bmatrix}$, $\begin{bmatrix}73&176\\94&237\end{bmatrix}$, $\begin{bmatrix}137&152\\180&133\end{bmatrix}$, $\begin{bmatrix}209&184\\123&239\end{bmatrix}$, $\begin{bmatrix}229&160\\102&197\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.dk.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: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 36x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^2}\cdot\frac{12921120x^{2}y^{12}z^{2}+5923931728896x^{2}y^{8}z^{6}-51499361360609280x^{2}y^{4}z^{10}+538990877234083921920x^{2}z^{14}-6192xy^{14}z+271155527424xy^{10}z^{5}-8840731536654336xy^{6}z^{9}+74881781010929811456xy^{2}z^{13}+y^{16}-9374683392y^{12}z^{4}-155483208695808y^{8}z^{8}+1645880657768349696y^{4}z^{12}+4738381338321616896z^{16}}{zy^{4}(468x^{2}y^{8}z+841487616x^{2}y^{4}z^{5}+19982861844480x^{2}z^{9}+xy^{10}+12503808xy^{6}z^{4}+1673945616384xy^{2}z^{8}+90720y^{8}z^{3}+31563343872y^{4}z^{7}+2821109907456z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.48.0-8.q.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-8.q.1.5 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 240.192.1-24.cv.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-24.cv.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-24.cv.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-24.cv.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-120.ql.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-120.ql.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-120.ql.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-120.ql.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-48.fj.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fj.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fn.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fn.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fx.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fx.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gd.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gd.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gg.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gg.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gj.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gj.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qu.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qu.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rc.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rc.1.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ro.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ro.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rr.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rr.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sx.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sx.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tb.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tb.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-24.ux.1.2 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-24.hz.1.10 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-120.nn.1.10 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |