Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | ||||
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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.0.708 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&14\\12&13\end{bmatrix}$, $\begin{bmatrix}7&4\\4&3\end{bmatrix}$, $\begin{bmatrix}13&20\\16&1\end{bmatrix}$, $\begin{bmatrix}15&4\\8&9\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
Contains $-I$: | no $\quad$ (see 24.48.0.m.2 for the level structure with $-I$) |
Cyclic 24-isogeny field degree: | $8$ |
Cyclic 24-torsion field degree: | $64$ |
Full 24-torsion field degree: | $768$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^2}\cdot\frac{x^{48}(6561x^{16}-139968x^{14}y^{2}+11757312x^{12}y^{4}-118444032x^{10}y^{6}+554065920x^{8}y^{8}-842268672x^{6}y^{10}+594542592x^{4}y^{12}-50331648x^{2}y^{14}+16777216y^{16})^{3}}{y^{4}x^{52}(3x^{2}-8y^{2})^{4}(3x^{2}+8y^{2})^{8}(9x^{4}-144x^{2}y^{2}+64y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-8.e.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.e.1.16 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.e.1.20 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.h.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.h.2.20 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.