Invariants
| Level: | $120$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| 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 | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| 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: | 20H1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}28&41\\41&78\end{bmatrix}$, $\begin{bmatrix}70&83\\21&32\end{bmatrix}$, $\begin{bmatrix}73&44\\96&91\end{bmatrix}$, $\begin{bmatrix}95&38\\74&109\end{bmatrix}$, $\begin{bmatrix}101&48\\100&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.1.l.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: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + y w $ |
| $=$ | $y^{2} - 2 y w - 3 z^{2} + 5 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 x^{2} z^{2} - 3 y^{2} z^{2} + 45 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{2916yz^{16}w-25920yz^{14}w^{3}+95472yz^{12}w^{5}-186624yz^{10}w^{7}+203520yz^{8}w^{9}-113152yz^{6}w^{11}+15360yz^{4}w^{13}+12288yz^{2}w^{15}-4096yw^{17}-729z^{18}-5832z^{16}w^{2}+103680z^{14}w^{4}-521856z^{12}w^{6}+1365120z^{10}w^{8}-2111232z^{8}w^{10}+1986304z^{6}w^{12}-1105920z^{4}w^{14}+331776z^{2}w^{16}-40960w^{18}}{w^{10}(3z^{2}-4w^{2})^{2}(6yz^{2}w-4yw^{3}-9z^{4}+39z^{2}w^{2}-40w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.1.l.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6X^{2}Z^{2}-3Y^{2}Z^{2}+45Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.72.1-20.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | 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.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.by.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ey.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fb.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fh.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fj.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ft.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fw.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fz.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.np.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bjt.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bko.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.blz.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bms.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqf.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bra.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-60.bd.1.29 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |