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: | 48.96.1.1570 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&31\\0&41\end{bmatrix}$, $\begin{bmatrix}7&46\\12&5\end{bmatrix}$, $\begin{bmatrix}23&0\\20&29\end{bmatrix}$, $\begin{bmatrix}43&31\\24&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bn.1 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 $ | $=$ | $ - 3 x w + z^{2} + z w + w^{2} $ |
| $=$ | $12 x^{2} - y^{2} + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 15 x^{2} z^{2} + 11 x z^{3} - 3 y^{2} z^{2} + 7 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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2}{3}\cdot\frac{18432xy^{10}w-472320xy^{8}w^{3}+4078080xy^{6}w^{5}-12581568xy^{4}w^{7}+7879032xy^{2}w^{9}-2985255xw^{11}-1024y^{12}+25344y^{10}w^{2}-213696y^{8}w^{4}+616896y^{6}w^{6}+28188y^{4}w^{8}+995085y^{2}w^{10}-746496w^{12}}{w^{2}y^{2}(480xy^{6}w+1152xy^{4}w^{3}+594xy^{2}w^{5}+81xw^{7}-32y^{8}-216y^{6}w^{2}-162y^{4}w^{4}-27y^{2}w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+8X^{3}Z+15X^{2}Z^{2}-3Y^{2}Z^{2}+11XZ^{3}+7Z^{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.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.2.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.2.26 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bz.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-16.b.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bd.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bo.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.co.2.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ct.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.df.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.di.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.fl.2.1 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bbe.1.10 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 96.192.5-96.bc.1.16 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bg.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bs.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.ce.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cq.1.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.dc.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.dg.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.dk.1.9 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.ia.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ie.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.iq.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.iu.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jq.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jy.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kw.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.le.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.db.2.6 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |