Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}11&38\\32&55\end{bmatrix}$, $\begin{bmatrix}23&90\\96&107\end{bmatrix}$, $\begin{bmatrix}35&63\\88&37\end{bmatrix}$, $\begin{bmatrix}41&8\\56&61\end{bmatrix}$, $\begin{bmatrix}47&105\\96&29\end{bmatrix}$, $\begin{bmatrix}79&7\\64&41\end{bmatrix}$, $\begin{bmatrix}79&33\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.dh.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $245760$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 16 y^{2} - z^{2} - w^{2} $ |
| $=$ | $x^{3} + y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 2 x^{2} y^{3} + z^{5} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1/4:0:1)$, $(0:1/4:1:0)$, $(0:1/4:0:1)$, $(0:-1/4:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{(z^{2}-4zw+w^{2})^{3}(z^{2}+4zw+w^{2})^{3}}{w^{2}z^{2}(z^{2}+w^{2})^{4}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.dh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y-\frac{1}{4}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}Y^{3}-X^{4}Z+Z^{5} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 40.48.0-8.q.1.4 | $40$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $3$ | $3$ | $0$ | $0$ |
| 120.72.2-12.p.1.4 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 120.72.2-12.p.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 120.72.2-24.cj.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 120.72.2-24.cj.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 120.72.2-24.cj.1.41 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 120.72.2-24.cj.1.43 | $120$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.