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: | 12D2 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}35&52\\88&61\end{bmatrix}$, $\begin{bmatrix}53&10\\34&109\end{bmatrix}$, $\begin{bmatrix}88&93\\57&34\end{bmatrix}$, $\begin{bmatrix}91&116\\8&15\end{bmatrix}$, $\begin{bmatrix}98&51\\41&88\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.36.2.y.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 $ | $=$ | $ x t + y t + z w $ |
| $=$ | $5 x w - z t$ | |
| $=$ | $5 x^{2} + 5 x y + z^{2}$ | |
| $=$ | $20 x^{2} - 20 x y + 20 y^{2} + 5 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} + x^{4} z^{2} + 15 x^{2} y^{2} z^{2} + 5 x^{2} z^{4} + 75 y^{2} z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} - 20x^{4} - 150x^{2} - 375 $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^2\,\frac{2880y^{2}z^{4}+3600y^{2}z^{2}t^{2}-780y^{2}t^{4}-624z^{4}t^{2}-456z^{2}t^{4}-125w^{6}-375w^{2}t^{4}-12t^{6}}{3840y^{2}z^{4}+480y^{2}z^{2}t^{2}-140y^{2}t^{4}+32z^{4}t^{2}+4z^{2}t^{4}-5w^{2}t^{4}-t^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.36.2.y.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{2}+X^{4}Z^{2}+15X^{2}Y^{2}Z^{2}+5X^{2}Z^{4}+75Y^{2}Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 60.36.2.y.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w^{2}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2zw^{6}t^{2}-\frac{6}{5}zw^{4}t^{4}-\frac{6}{25}zw^{2}t^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}wt^{2}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.36.1-12.b.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 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.v.1.12 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.cf.1.5 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.ej.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.ek.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.fp.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.fq.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.fx.1.14 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-60.fy.1.11 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.ge.1.7 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.oh.1.6 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bcb.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bci.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bjl.1.3 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.bjs.1.1 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.blp.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3-120.blw.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.360.14-60.bs.1.3 | $120$ | $5$ | $5$ | $14$ |
| 120.432.15-60.cq.1.13 | $120$ | $6$ | $6$ | $15$ |