Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.0.153 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&38\\44&7\end{bmatrix}$, $\begin{bmatrix}19&19\\20&31\end{bmatrix}$, $\begin{bmatrix}23&7\\32&17\end{bmatrix}$, $\begin{bmatrix}33&11\\4&21\end{bmatrix}$, $\begin{bmatrix}35&37\\24&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $24576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 138 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{x^{24}(x^{8}+4x^{6}y^{2}-10x^{4}y^{4}-28x^{2}y^{6}+y^{8})^{3}}{y^{4}x^{26}(x^{2}+y^{2})^{8}(x^{2}+2y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.24.0-8.n.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.24.0-8.n.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 48.96.0-8.j.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.m.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.n.2.4 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.o.1.3 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.u.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.w.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.y.1.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.ba.1.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.be.1.14 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bg.1.12 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bi.2.3 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bk.1.4 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bm.1.3 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bm.2.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bo.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bo.1.12 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-16.q.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.s.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.u.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.w.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bq.1.12 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bs.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.by.1.12 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.ca.1.14 | $48$ | $2$ | $2$ | $1$ |
| 48.144.4-24.ge.1.17 | $48$ | $3$ | $3$ | $4$ |
| 48.192.3-24.gf.2.30 | $48$ | $4$ | $4$ | $3$ |
| 240.96.0-40.bi.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bk.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bm.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bm.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bo.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bo.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bu.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bw.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.co.1.26 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cq.2.26 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.de.2.22 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.dg.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.ed.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.eh.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.el.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.ep.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-80.bs.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.bu.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.ca.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.cc.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fo.1.30 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fq.2.30 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ge.2.30 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gg.1.30 | $240$ | $2$ | $2$ | $1$ |
| 240.240.8-40.da.2.1 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-40.fn.2.31 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-40.gq.2.20 | $240$ | $10$ | $10$ | $15$ |