Invariants
| Level: | $120$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $20^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 14$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A8 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&66\\102&55\end{bmatrix}$, $\begin{bmatrix}49&84\\28&55\end{bmatrix}$, $\begin{bmatrix}59&56\\40&53\end{bmatrix}$, $\begin{bmatrix}87&38\\2&25\end{bmatrix}$, $\begin{bmatrix}101&36\\72&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $147456$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t + x v + y t - y v - w r $ |
| $=$ | $x r + y u + w t - w v$ | |
| $=$ | $x t - x v - y v + z r - w r$ | |
| $=$ | $x u - x r - 2 y r + w v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 32 x^{10} - 16 x^{8} y^{2} + 32 x^{8} z^{2} + 2 x^{6} y^{4} - 44 x^{6} y^{2} z^{2} + 8 x^{6} z^{4} + \cdots + 8 y^{2} z^{8} $ |
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
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 20.60.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}+XY+2Y^{2}+ZW+W^{2} $ |
| $=$ | $ X^{3}-X^{2}Y-XZ^{2}-XZW-YZW+XW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 32X^{10}-16X^{8}Y^{2}+32X^{8}Z^{2}+2X^{6}Y^{4}-44X^{6}Y^{2}Z^{2}+8X^{6}Z^{4}-X^{4}Y^{4}Z^{2}+2X^{4}Y^{2}Z^{4}+6X^{2}Y^{6}Z^{2}+8X^{2}Y^{4}Z^{4}+24X^{2}Y^{2}Z^{6}-Y^{8}Z^{2}+6Y^{6}Z^{4}-12Y^{4}Z^{6}+8Y^{2}Z^{8} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 24.48.0-8.a.1.3 | $24$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-8.a.1.3 | $24$ | $5$ | $5$ | $0$ | $0$ |
| 60.120.4-20.a.1.4 | $60$ | $2$ | $2$ | $4$ | $1$ |
| 120.120.4-20.a.1.2 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-40.a.1.1 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-40.a.1.2 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 120.120.4-40.a.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.480.17-40.e.1.1 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-120.e.1.7 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-40.f.1.1 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-120.f.1.8 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-40.g.1.3 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-120.g.1.7 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-40.h.1.3 | $120$ | $2$ | $2$ | $17$ |
| 120.480.17-120.h.1.8 | $120$ | $2$ | $2$ | $17$ |
| 240.480.16-80.a.1.2 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-240.a.1.7 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-80.b.1.2 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-240.b.1.7 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-80.c.1.2 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-240.c.1.7 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-80.d.1.2 | $240$ | $2$ | $2$ | $16$ |
| 240.480.16-240.d.1.7 | $240$ | $2$ | $2$ | $16$ |
| 240.480.18-80.a.1.8 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-240.a.1.7 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-80.b.1.8 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-240.b.1.6 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-80.c.1.8 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-240.c.1.6 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-80.d.1.8 | $240$ | $2$ | $2$ | $18$ |
| 240.480.18-240.d.1.4 | $240$ | $2$ | $2$ | $18$ |