Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16H0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.103 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}41&18\\0&13\end{bmatrix}$, $\begin{bmatrix}41&36\\40&25\end{bmatrix}$, $\begin{bmatrix}47&7\\20&3\end{bmatrix}$, $\begin{bmatrix}47&29\\24&29\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.0.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: | $12288$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 3 x^{2} - 6 y^{2} + 4 z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.48.0-8.bb.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.h.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.h.1.15 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-8.bb.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
48.192.1-48.f.1.12 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.z.2.5 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.bi.2.4 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.by.1.5 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.ci.1.8 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cs.2.10 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cw.2.6 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dk.1.7 | $48$ | $2$ | $2$ | $1$ |
48.288.8-48.if.2.3 | $48$ | $3$ | $3$ | $8$ |
48.384.7-48.ha.2.15 | $48$ | $4$ | $4$ | $7$ |
240.192.1-240.xe.1.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.xm.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.yk.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ys.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zq.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zy.2.11 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.baw.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bbe.2.13 | $240$ | $2$ | $2$ | $1$ |
240.480.16-240.fz.2.22 | $240$ | $5$ | $5$ | $16$ |