Invariants
| Level: | $48$ | $\SL_2$-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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1565 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&1\\4&29\end{bmatrix}$, $\begin{bmatrix}5&26\\32&9\end{bmatrix}$, $\begin{bmatrix}25&23\\12&5\end{bmatrix}$, $\begin{bmatrix}35&3\\32&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bl.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 24 x y + 6 y^{2} + w^{2} $ |
| $=$ | $24 x^{2} - 6 x y + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 3 x^{2} y^{2} + 9 x^{2} z^{2} + 18 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{387072y^{2}z^{10}+13824y^{2}z^{8}w^{2}-5453568y^{2}z^{6}w^{4}-24107328y^{2}z^{4}w^{6}-18878616y^{2}z^{2}w^{8}-1572858y^{2}w^{10}+131072z^{12}+196608z^{10}w^{2}-303360z^{8}w^{4}-191744z^{6}w^{6}+173568z^{4}w^{8}-1049280z^{2}w^{10}-131071w^{12}}{w^{2}z^{2}(384y^{2}z^{6}+1056y^{2}z^{4}w^{2}+168y^{2}z^{2}w^{4}+6y^{2}w^{6}+512z^{6}w^{2}+272z^{4}w^{4}+32z^{2}w^{6}+w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bl.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+3X^{2}Y^{2}+9X^{2}Z^{2}+18Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.b.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.bz.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.e.2.26 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bz.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-16.b.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.1-48.l.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bc.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bk.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cc.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cq.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cr.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dd.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dk.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.fb.1.1 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bbc.2.4 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 96.192.5-96.ba.2.16 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.be.2.12 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bq.2.12 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cc.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.co.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.da.2.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.de.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.di.2.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.hy.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ic.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.io.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.is.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jo.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jw.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ku.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.lc.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.cv.2.4 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |