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 $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.0.126 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}19&7\\36&43\end{bmatrix}$, $\begin{bmatrix}19&34\\16&11\end{bmatrix}$, $\begin{bmatrix}21&14\\4&19\end{bmatrix}$, $\begin{bmatrix}35&30\\24&19\end{bmatrix}$, $\begin{bmatrix}43&45\\4&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.bb.2 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 221 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(x-y)^{24}(193x^{8}-2784x^{7}y+11384x^{6}y^{2}+12352x^{5}y^{3}-204840x^{4}y^{4}+529792x^{3}y^{5}-289056x^{2}y^{6}-929024xy^{7}+1262608y^{8})^{3}}{(x-6y)^{2}(x-y)^{28}(x+4y)(3x-8y)(x^{2}+8xy-34y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.24.0-8.n.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.24.0-8.n.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.l.1.4 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.m.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.n.1.4 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-8.p.1.1 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.v.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.x.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.z.2.5 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.bb.2.5 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-16.r.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.t.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.v.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.x.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.96.0-24.bj.1.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bl.2.6 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bn.1.5 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bp.2.8 | $48$ | $2$ | $2$ | $0$ |
| 48.144.4-24.gl.2.12 | $48$ | $3$ | $3$ | $4$ |
| 48.192.3-24.gi.2.7 | $48$ | $4$ | $4$ | $3$ |
| 240.96.0-40.bj.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bl.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bn.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-40.bp.2.8 | $240$ | $2$ | $2$ | $0$ |
| 240.240.8-40.dd.2.14 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-40.fs.2.6 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-40.gx.1.9 | $240$ | $10$ | $10$ | $15$ |
| 48.96.0-48.bf.2.13 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bh.2.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bn.1.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bp.2.13 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.br.2.13 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bt.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bz.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.cb.2.13 | $48$ | $2$ | $2$ | $1$ |
| 240.96.0-80.bn.2.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bp.2.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bv.1.7 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bx.2.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-80.bt.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.bv.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.cb.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.cd.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.0-120.eg.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.ek.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.eo.1.12 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.es.2.14 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cp.2.14 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cr.2.15 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.df.2.15 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.dh.2.15 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.fp.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fr.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gf.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gh.2.6 | $240$ | $2$ | $2$ | $1$ |