Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| 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: | $\begin{bmatrix}31&208\\59&153\end{bmatrix}$, $\begin{bmatrix}53&112\\143&51\end{bmatrix}$, $\begin{bmatrix}57&104\\98&157\end{bmatrix}$, $\begin{bmatrix}125&192\\224&161\end{bmatrix}$, $\begin{bmatrix}193&72\\117&35\end{bmatrix}$, $\begin{bmatrix}221&136\\39&211\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.h.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: | 288.2.a.d |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2}\cdot\frac{16187040x^{2}y^{12}z^{2}-962206179757056x^{2}y^{8}z^{6}+319479483107509862400x^{2}y^{4}z^{10}-2208245624028041828106240x^{2}z^{14}-6768xy^{14}z+5634505338624xy^{10}z^{5}-8217092592118923264xy^{6}z^{9}+306700803051956449837056xy^{2}z^{13}+y^{16}-15651595008y^{12}z^{4}+91326841956974592y^{8}z^{8}-6815591803818736091136y^{4}z^{12}+4738381338321616896z^{16}}{zy^{4}(36x^{2}y^{8}z-1679616x^{2}y^{4}z^{5}-78364164096x^{2}z^{9}-xy^{10}-2176782336xy^{2}z^{8}+2592y^{8}z^{3}-120932352y^{4}z^{7}-2821109907456z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 240.48.0-48.g.1.8 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-48.g.1.27 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-8.q.1.4 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-48.b.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-48.b.1.25 | $240$ | $2$ | $2$ | $1$ | $?$ | 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.bg.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bg.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bj.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bj.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bn.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bn.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bq.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bq.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dw.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dw.2.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ed.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ed.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.el.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.el.2.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.eq.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.eq.2.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-48.fp.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fs.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fs.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fv.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fw.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gb.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gb.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gd.1.15 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rw.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ry.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ry.2.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rz.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.si.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sm.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sm.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.so.1.29 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-48.be.1.14 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.mk.1.42 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.p.1.38 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |