Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 36$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}34&1\\91&32\end{bmatrix}$, $\begin{bmatrix}57&112\\112&93\end{bmatrix}$, $\begin{bmatrix}66&31\\43&36\end{bmatrix}$, $\begin{bmatrix}69&26\\62&69\end{bmatrix}$, $\begin{bmatrix}105&64\\116&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.1.bb.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $491520$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y - 6 y^{2} + w^{2} $ |
| $=$ | $6 x^{2} + 6 x y + 6 y^{2} + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{2} y^{2} - 6 x^{2} z^{2} + 12 z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{192xz^{8}-577xz^{6}w^{2}-27xz^{4}w^{4}+87xz^{2}w^{6}+9xw^{8}-191yz^{8}+25yz^{6}w^{2}+288yz^{4}w^{4}+75yz^{2}w^{6}}{xz^{6}w^{2}-6xz^{2}w^{6}-9xw^{8}-yz^{8}+2yz^{6}w^{2}+9yz^{4}w^{4}+6yz^{2}w^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.1.bb.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{2}Y^{2}-6X^{2}Z^{2}+12Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.36.1-12.b.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1-12.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 |
|---|---|---|---|---|---|---|
| 120.144.3-24.h.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ck.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.fj.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.fn.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jj.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jl.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jx.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jz.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bhf.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bhh.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bht.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bhv.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjj.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjl.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjx.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjz.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.360.13-120.cb.1.15 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.432.13-120.et.1.29 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |