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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.727 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&28\\28&15\end{bmatrix}$, $\begin{bmatrix}21&22\\16&41\end{bmatrix}$, $\begin{bmatrix}27&20\\4&1\end{bmatrix}$, $\begin{bmatrix}37&9\\24&43\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.0.w.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $32$ |
Full 48-torsion field degree: | $12288$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 3 x^{2} - 12 x z + 2 y^{2} - 12 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 |
---|---|---|---|---|---|
16.48.0-16.e.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bj.1.10 | $24$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-16.e.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bj.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cj.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.ck.1.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cr.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cs.1.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dp.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dq.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dx.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dy.2.1 | $48$ | $2$ | $2$ | $1$ |
48.288.8-48.dg.1.1 | $48$ | $3$ | $3$ | $8$ |
48.384.7-48.ee.2.17 | $48$ | $4$ | $4$ | $7$ |
240.192.1-240.lz.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ma.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.mh.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.mi.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qx.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qy.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rf.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rg.2.1 | $240$ | $2$ | $2$ | $1$ |
240.480.16-240.by.1.2 | $240$ | $5$ | $5$ | $16$ |