Invariants
| Level: | $120$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 72$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}43&26\\96&13\end{bmatrix}$, $\begin{bmatrix}85&112\\54&7\end{bmatrix}$, $\begin{bmatrix}93&32\\58&69\end{bmatrix}$, $\begin{bmatrix}103&30\\104&107\end{bmatrix}$, $\begin{bmatrix}107&88\\102&1\end{bmatrix}$, $\begin{bmatrix}109&92\\8&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.1.b.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $16$ |
| Cyclic 120-torsion field degree: | $512$ |
| Full 120-torsion field degree: | $245760$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 15 x^{2} - z^{2} - 4 z w $ |
| $=$ | $15 x y + 15 y^{2} + z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} - 15 x^{2} y^{2} - 30 x^{2} y z - 30 x^{2} z^{2} + y^{2} z^{2} - 2 y z^{3} + z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{6}+4z^{5}w+240z^{4}w^{2}+480z^{3}w^{3}+1440z^{2}w^{4}+944zw^{5}+16w^{6})^{3}}{w^{2}z(z-w)^{10}(z+4w)^{5}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.1.b.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 225X^{4}-15X^{2}Y^{2}-30X^{2}YZ-30X^{2}Z^{2}+Y^{2}Z^{2}-2YZ^{3}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.72.1-10.a.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.288.5-60.be.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.be.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bf.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bf.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bf.2.35 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bi.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bi.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bi.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bj.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bj.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bj.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bq.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bq.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bq.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.br.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.br.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.br.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bu.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bu.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bu.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bv.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bv.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bv.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dm.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dm.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dm.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dp.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dp.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dp.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dy.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dy.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dy.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eb.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eb.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eb.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ew.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ew.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ew.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ez.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ez.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ez.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fi.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fi.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fi.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fl.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fl.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fl.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-60.b.2.4 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |