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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.0.340 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&28\\40&23\end{bmatrix}$, $\begin{bmatrix}13&10\\24&7\end{bmatrix}$, $\begin{bmatrix}27&8\\40&39\end{bmatrix}$, $\begin{bmatrix}31&14\\0&13\end{bmatrix}$, $\begin{bmatrix}45&29\\32&33\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.24.0.bh.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
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 54 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3}\cdot\frac{(3x+y)^{24}(9x^{4}-36x^{3}y-18x^{2}y^{2}+12xy^{3}+y^{4})^{3}(9x^{4}+36x^{3}y-18x^{2}y^{2}-12xy^{3}+y^{4})^{3}}{y^{2}x^{2}(3x+y)^{24}(3x^{2}-y^{2})^{2}(3x^{2}+y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
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.