Invariants
Level: | $120$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
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: | 8C0 |
Level structure
$\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}28&41\\115&42\end{bmatrix}$, $\begin{bmatrix}41&18\\76&91\end{bmatrix}$, $\begin{bmatrix}57&92\\26&103\end{bmatrix}$, $\begin{bmatrix}92&77\\69&112\end{bmatrix}$, $\begin{bmatrix}110&103\\51&14\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.12.0.o.1 for the level structure with $-I$) |
Cyclic 120-isogeny field degree: | $48$ |
Cyclic 120-torsion field degree: | $1536$ |
Full 120-torsion field degree: | $1474560$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1491 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}+16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x^{2}+16y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
60.12.0-4.c.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
120.12.0-4.c.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
120.48.0-8.i.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-8.j.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-8.w.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-8.y.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bo.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bq.1.8 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bs.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bs.1.7 | $120$ | $2$ | $2$ | $0$ |
120.48.0-24.bu.1.8 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bu.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.bw.1.7 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.by.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dw.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.dy.1.16 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ee.1.10 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.eg.1.16 | $120$ | $2$ | $2$ | $0$ |
120.72.2-24.cw.1.31 | $120$ | $3$ | $3$ | $2$ |
120.96.1-24.iw.1.17 | $120$ | $4$ | $4$ | $1$ |
120.120.4-40.bq.1.16 | $120$ | $5$ | $5$ | $4$ |
120.144.3-40.cg.1.28 | $120$ | $6$ | $6$ | $3$ |
120.240.7-40.cw.1.19 | $120$ | $10$ | $10$ | $7$ |
240.48.0-16.i.1.4 | $240$ | $2$ | $2$ | $0$ |
240.48.0-48.i.1.3 | $240$ | $2$ | $2$ | $0$ |
240.48.0-16.j.1.4 | $240$ | $2$ | $2$ | $0$ |
240.48.0-48.j.1.4 | $240$ | $2$ | $2$ | $0$ |
240.48.0-80.q.1.6 | $240$ | $2$ | $2$ | $0$ |
240.48.0-240.q.1.17 | $240$ | $2$ | $2$ | $0$ |
240.48.0-80.r.1.2 | $240$ | $2$ | $2$ | $0$ |
240.48.0-240.r.1.19 | $240$ | $2$ | $2$ | $0$ |
240.48.1-16.c.1.5 | $240$ | $2$ | $2$ | $1$ |
240.48.1-48.c.1.13 | $240$ | $2$ | $2$ | $1$ |
240.48.1-80.c.1.15 | $240$ | $2$ | $2$ | $1$ |
240.48.1-240.c.1.12 | $240$ | $2$ | $2$ | $1$ |
240.48.1-16.d.1.5 | $240$ | $2$ | $2$ | $1$ |
240.48.1-48.d.1.14 | $240$ | $2$ | $2$ | $1$ |
240.48.1-80.d.1.11 | $240$ | $2$ | $2$ | $1$ |
240.48.1-240.d.1.16 | $240$ | $2$ | $2$ | $1$ |