Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}23&32\\6&55\end{bmatrix}$, $\begin{bmatrix}23&84\\36&95\end{bmatrix}$, $\begin{bmatrix}25&96\\102&37\end{bmatrix}$, $\begin{bmatrix}71&58\\58&81\end{bmatrix}$, $\begin{bmatrix}85&18\\6&91\end{bmatrix}$, $\begin{bmatrix}103&110\\74&21\end{bmatrix}$, $\begin{bmatrix}107&104\\50&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.4.b.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $245760$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 5 x^{2} - z^{2} - w^{2} $ |
| $=$ | $ - x z w + 20 y^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 12 x^{5} z + 25 x^{4} z^{2} - 10 x^{3} y^{3} + 30 x^{2} y^{3} z - 125 x^{2} z^{4} + \cdots - 125 z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.72.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-\frac{1}{2}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}z-\frac{1}{10}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-10X^{3}Y^{3}-12X^{5}Z+30X^{2}Y^{3}Z+25X^{4}Z^{2}-30XY^{3}Z^{2}+10Y^{3}Z^{3}-125X^{2}Z^{4}+300XZ^{5}-125Z^{6} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 40.48.0-20.b.1.9 | $40$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.2-12.a.1.14 | $24$ | $2$ | $2$ | $2$ | $0$ |
| 40.48.0-20.b.1.9 | $40$ | $3$ | $3$ | $0$ | $0$ |
| 120.72.2-12.a.1.19 | $120$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.