Invariants
| Level: | $120$ | $\SL_2$-level: | $40$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J3 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}47&30\\74&101\end{bmatrix}$, $\begin{bmatrix}53&80\\21&97\end{bmatrix}$, $\begin{bmatrix}73&50\\108&83\end{bmatrix}$, $\begin{bmatrix}91&110\\18&37\end{bmatrix}$, $\begin{bmatrix}119&110\\28&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.3.fd.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $16$ |
| Cyclic 120-torsion field degree: | $512$ |
| Full 120-torsion field degree: | $245760$ |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x w + 2 y z $ |
| $=$ | $z u + 2 w t - 2 w u$ | |
| $=$ | $x^{2} + 4 y^{2} + z^{2} - 2 z w$ | |
| $=$ | $x u + 4 y t - 4 y u$ | |
| $=$ | $\cdots$ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 25 w^{2} $ | $=$ | $ -225 x^{4} + 15 x^{2} z^{2} + z^{4} $ |
| $0$ | $=$ | $-15 x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3\,\frac{10800000z^{8}t^{2}+10800000z^{8}tu+5760000z^{6}t^{2}u^{2}-3240000z^{6}tu^{3}+630000z^{6}u^{4}-474000z^{4}t^{2}u^{4}+783000z^{4}tu^{5}+104625z^{4}u^{6}-66700z^{2}t^{2}u^{6}+110660z^{2}tu^{7}-79985z^{2}u^{8}-118125w^{8}u^{2}+1125w^{6}u^{4}+21900w^{4}u^{6}+44085w^{2}u^{8}+364t^{2}u^{8}+424tu^{9}-409u^{10}}{u^{4}(540000z^{6}-288000z^{4}t^{2}+468000z^{4}tu-63000z^{4}u^{2}-19800z^{2}t^{2}u^{2}+44100z^{2}tu^{3}-20955z^{2}u^{4}-6750w^{6}-1575w^{4}u^{2}-180w^{2}u^{4}-44t^{2}u^{4}+216tu^{5}-151u^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ |
| 120.24.0-60.f.1.8 | $120$ | $6$ | $6$ | $0$ | $?$ |
| 120.72.1-20.b.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.288.5-60.fq.1.11 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fq.2.9 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fr.1.6 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fr.2.5 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fy.1.5 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fy.2.1 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fz.1.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-60.fz.2.1 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.boo.1.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.boo.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bov.1.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bov.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bqs.1.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bqs.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bqz.1.3 | $120$ | $2$ | $2$ | $5$ |
| 120.288.5-120.bqz.2.2 | $120$ | $2$ | $2$ | $5$ |
| 120.432.15-60.cf.1.5 | $120$ | $3$ | $3$ | $15$ |