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 $2$ are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
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: | 16G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.351 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&8\\32&11\end{bmatrix}$, $\begin{bmatrix}3&32\\40&19\end{bmatrix}$, $\begin{bmatrix}29&22\\16&27\end{bmatrix}$, $\begin{bmatrix}37&34\\32&13\end{bmatrix}$, $\begin{bmatrix}39&44\\8&5\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.0.d.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}\cdot3^8}\cdot\frac{(x-2y)^{48}(6561x^{16}-331776x^{8}y^{8}+16777216y^{16})^{3}}{y^{16}x^{16}(x-2y)^{48}(3x^{2}-8y^{2})^{2}(3x^{2}+8y^{2})^{2}(9x^{4}+64y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.0-8.i.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-8.i.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.e.1.32 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.f.2.24 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.