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: | $\begin{bmatrix}9&32\\104&81\end{bmatrix}$, $\begin{bmatrix}47&16\\8&77\end{bmatrix}$, $\begin{bmatrix}80&41\\67&26\end{bmatrix}$, $\begin{bmatrix}94&77\\1&100\end{bmatrix}$, $\begin{bmatrix}99&4\\28&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.1.bn.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 $ | $=$ | $ 2 x y - 2 y^{2} - w^{2} $ |
| $=$ | $2 x^{2} + 2 x y + 2 y^{2} - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 2 x^{2} y^{2} + 6 x^{2} z^{2} + 4 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 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.bn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}-2X^{2}Y^{2}+6X^{2}Z^{2}+4Z^{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.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | 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.bb.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.cg.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.fc.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.fg.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.lg.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.li.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.mb.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.md.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bln.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.blp.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bmb.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bmd.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bnr.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bnt.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bof.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.boh.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.360.13-120.cn.1.7 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.432.13-120.ff.1.11 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |