Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}5&37\\72&37\end{bmatrix}$, $\begin{bmatrix}5&64\\38&87\end{bmatrix}$, $\begin{bmatrix}27&26\\82&29\end{bmatrix}$, $\begin{bmatrix}41&87\\96&53\end{bmatrix}$, $\begin{bmatrix}119&106\\12&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.ie.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $30 x^{2} + 6 y^{2} + 30 y z + 54 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 10 x^{2} z^{2} - 6 y^{2} z^{2} + 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{17390370816yz^{11}+3901685760yz^{9}w^{2}+304570368yz^{7}w^{4}+9234432yz^{5}w^{6}+75456yz^{3}w^{8}+144yzw^{10}+17199267840z^{12}+3551330304z^{10}w^{2}+223948800z^{8}w^{4}+3096576z^{6}w^{6}-101376z^{4}w^{8}-1008z^{2}w^{10}-w^{12}}{w^{2}z^{6}(23328yz^{3}+108yzw^{2}+23328z^{4}-378z^{2}w^{2}-w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ie.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+10X^{2}Z^{2}-6Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.24.0-24.m.1.2 | $120$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.f.1.9 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.f.1.10 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.192.3-24.fi.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fi.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fi.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fi.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fj.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fj.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fj.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fj.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lo.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lo.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lo.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lo.2.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lp.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lp.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lp.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lp.2.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-24.fv.1.4 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-120.bpq.1.19 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |