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: | $10$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.78 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&28\\40&39\end{bmatrix}$, $\begin{bmatrix}5&44\\0&47\end{bmatrix}$, $\begin{bmatrix}7&6\\40&41\end{bmatrix}$, $\begin{bmatrix}9&28\\8&7\end{bmatrix}$, $\begin{bmatrix}15&14\\8&17\end{bmatrix}$, $\begin{bmatrix}29&2\\40&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.l.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $12288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{(x+2y)^{48}(x^{16}+32x^{15}y+1408x^{14}y^{2}+30464x^{13}y^{3}+378624x^{12}y^{4}+3123200x^{11}y^{5}+18391040x^{10}y^{6}+80003072x^{9}y^{7}+260857856x^{8}y^{8}+640024576x^{7}y^{9}+1177026560x^{6}y^{10}+1599078400x^{5}y^{11}+1550843904x^{4}y^{12}+998244352x^{3}y^{13}+369098752x^{2}y^{14}+67108864xy^{15}+16777216y^{16})^{3}}{y^{2}x^{2}(x+2y)^{50}(x+4y)^{2}(x^{2}-8y^{2})^{8}(x^{2}+4xy+8y^{2})^{4}(x^{2}+8xy+8y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.48.0-8.i.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.i.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-8.bb.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.