Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.141 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&6\\32&17\end{bmatrix}$, $\begin{bmatrix}13&24\\0&37\end{bmatrix}$, $\begin{bmatrix}25&43\\28&11\end{bmatrix}$, $\begin{bmatrix}33&25\\34&31\end{bmatrix}$, $\begin{bmatrix}35&16\\2&21\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| Full 48-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y^{2} - 4 y w + 2 z^{2} + 2 w^{2} $ |
| $=$ | $6 x^{2} + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} + 2 y^{2} z^{2} - 4 y z^{3} + 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{432yz^{10}w-25616yz^{8}w^{3}+300416yz^{6}w^{5}-1170816yz^{4}w^{7}+1761024yz^{2}w^{9}-887040yw^{11}-27z^{12}+5256z^{10}w^{2}-97100z^{8}w^{4}+419008z^{6}w^{6}-402576z^{4}w^{8}-404352z^{2}w^{10}+519616w^{12}}{z^{8}(4yz^{2}w-12yw^{3}-z^{4}+6z^{2}w^{2}+7w^{4})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bq.1 | $8$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.96.3.bb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.en.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.iy.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.jh.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tu.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tv.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.ve.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.vf.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.xl.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.xm.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.xt.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.xu.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.zn.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.zo.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.zr.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.zs.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.5.lt.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.lu.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.lx.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.ly.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.qp.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.qq.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.qx.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.qy.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.144.7.sy.1 | $48$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 48.192.11.lw.1 | $48$ | $4$ | $4$ | $11$ | $2$ | $1^{10}$ |
| 240.96.3.ezs.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ezy.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fbc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fbi.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fin.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fiz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.flh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.flt.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fpz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fqa.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fqp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fqq.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fub.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fuc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fuj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fuk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.din.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dio.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.div.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.diw.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dnb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dnc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dnr.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dns.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.240.17.bav.1 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 240.288.17.xer.1 | $240$ | $6$ | $6$ | $17$ | $?$ | not computed |