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 $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}5&24\\184&205\end{bmatrix}$, $\begin{bmatrix}119&56\\190&167\end{bmatrix}$, $\begin{bmatrix}119&72\\196&179\end{bmatrix}$, $\begin{bmatrix}165&16\\146&157\end{bmatrix}$, $\begin{bmatrix}165&232\\176&217\end{bmatrix}$, $\begin{bmatrix}175&8\\131&141\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.g.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} + 9x $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^2}\cdot\frac{1011690x^{2}y^{12}z^{2}-939654472419x^{2}y^{8}z^{6}+4874870042534025x^{2}y^{4}z^{10}-526486783988008935x^{2}z^{14}-1692xy^{14}z+22009786479xy^{10}z^{5}-501531530280696xy^{6}z^{9}+292492678691822481xy^{2}z^{13}+y^{16}-244556172y^{12}z^{4}+22296592274652y^{8}z^{8}-25999419417643494y^{4}z^{12}+282429536481z^{16}}{zy^{4}(9x^{2}y^{8}z-6561x^{2}y^{4}z^{5}-4782969x^{2}z^{9}-xy^{10}-531441xy^{2}z^{8}+162y^{8}z^{3}-118098y^{4}z^{7}-43046721z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 240.48.0-48.h.1.6 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.h.1.28 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-8.q.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-48.a.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-48.a.1.27 | $240$ | $2$ | $2$ | $1$ | $?$ | 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.bf.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bf.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bi.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bi.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bm.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bm.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bp.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bp.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dt.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dt.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dy.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dy.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.eg.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.eg.2.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.en.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.en.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-48.fp.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fq.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fq.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fr.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fx.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fz.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fz.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.ga.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rs.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rt.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rt.2.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ru.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sd.1.25 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.se.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.se.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sf.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-48.z.1.6 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.mh.1.34 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.m.1.44 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |