Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $3600$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}43&31\\22&47\end{bmatrix}$, $\begin{bmatrix}57&59\\80&37\end{bmatrix}$, $\begin{bmatrix}63&19\\28&107\end{bmatrix}$, $\begin{bmatrix}71&119\\8&79\end{bmatrix}$, $\begin{bmatrix}75&58\\26&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.36.2.v.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $491520$ |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 4 x y - z t $ |
| $=$ | $15 x^{2} - y t$ | |
| $=$ | $ - 15 x z + 4 y^{2}$ | |
| $=$ | $60 z^{2} + 15 w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3375 x^{6} - y^{2} z^{4} - z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} + 3375 $ |
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 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{(5w^{2}+t^{2})^{3}}{t^{2}(15w^{2}-t^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.36.2.v.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{15}{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{15}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3375X^{6}-Y^{2}Z^{4}-Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.36.2.v.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{16}z^{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{15}y$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.36.1-12.b.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 120.24.0-60.f.1.8 | $120$ | $3$ | $3$ | $0$ | $?$ |
| 120.36.1-12.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.144.3-60.fq.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.fr.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gg.1.10 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gh.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gw.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.gx.1.1 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.hm.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.hn.1.7 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bjq.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bjx.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bny.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bof.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bsg.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bsn.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwo.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bwv.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.360.14-60.bp.1.6 | $120$ | $5$ | $5$ | $14$ |
| 120.432.15-60.cf.1.5 | $120$ | $6$ | $6$ | $15$ |