Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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: | $\begin{bmatrix}19&84\\78&85\end{bmatrix}$, $\begin{bmatrix}33&43\\10&57\end{bmatrix}$, $\begin{bmatrix}77&13\\6&85\end{bmatrix}$, $\begin{bmatrix}79&59\\102&119\end{bmatrix}$, $\begin{bmatrix}83&31\\54&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.i.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: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $4 x^{2} + 7 y^{2} + 4 y z + y w + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{4} + x^{2} y z + 8 x^{2} z^{2} + y^{2} z^{2} + y z^{3} + 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{7^2}\cdot\frac{9448990857216yz^{11}+1574903079936yz^{10}w-10236058642944yz^{9}w^{2}-2842888534272yz^{8}w^{3}+2762569259136yz^{7}w^{4}+904262885568yz^{6}w^{5}-204328601568yz^{5}w^{6}-86704187184yz^{4}w^{7}+3203128800yz^{3}w^{8}+2866505760yz^{2}w^{9}+54686664yzw^{10}-25019280yw^{11}+1349860517888z^{12}+1574903079936z^{11}w+337805056512z^{10}w^{2}-1343061764352z^{9}w^{3}-1324551125376z^{8}w^{4}-18659654208z^{7}w^{5}+298304772192z^{6}w^{6}+52388950608z^{5}w^{7}-16408031328z^{4}w^{8}-4263770448z^{3}w^{9}+82382832z^{2}w^{10}+85917024zw^{11}+6205977w^{12}}{z^{2}(72yz^{9}+732yz^{8}w+3642yz^{7}w^{2}+12075yz^{6}w^{3}+29946yz^{5}w^{4}+59052yz^{4}w^{5}-1647114yz^{3}w^{6}-1027437yz^{2}w^{7}+57474yzw^{8}+55071yw^{9}+72z^{10}+732z^{9}w+3630z^{8}w^{2}+11951z^{7}w^{3}+29316z^{6}w^{4}+56910z^{5}w^{5}-158011z^{4}w^{6}-291345z^{3}w^{7}-324915z^{2}w^{8}-118387zw^{9}-3078w^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.48.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 7X^{4}+X^{2}YZ+8X^{2}Z^{2}+Y^{2}Z^{2}+YZ^{3}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.24.0-12.e.1.1 | $120$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.f.1.9 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.f.1.13 | $120$ | $2$ | $2$ | $0$ | $?$ | 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.fg.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fg.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fg.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fg.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fh.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fh.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fh.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.fh.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lm.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lm.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lm.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lm.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ln.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ln.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ln.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ln.2.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-12.r.1.3 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.q.1.15 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |