Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}15&16\\8&67\end{bmatrix}$, $\begin{bmatrix}23&24\\72&13\end{bmatrix}$, $\begin{bmatrix}79&28\\104&61\end{bmatrix}$, $\begin{bmatrix}81&92\\68&47\end{bmatrix}$, $\begin{bmatrix}95&16\\32&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.w.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $184320$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 x y - z^{2} $ |
| $=$ | $x^{2} + 2 x y - 5 y^{2} + 4 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 10 x^{2} y^{2} - 6 x^{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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{5^2}\cdot\frac{(625z^{8}-500z^{6}w^{2}+125z^{4}w^{4}-10z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(5z^{2}-2w^{2})^{2}(5z^{2}-w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+10X^{2}Y^{2}-6X^{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 |
|---|---|---|---|---|---|---|
| 24.96.0-8.c.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 120.96.0-40.b.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-40.b.1.13 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-8.c.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-40.s.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-40.s.1.9 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-40.t.1.6 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-40.t.1.12 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.1-40.n.2.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-40.n.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-40.bi.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-40.bi.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-40.bj.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-40.bj.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.384.5-40.w.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-40.y.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-40.z.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-40.bb.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hh.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hj.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hq.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ht.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.h.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.j.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.bf.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bf.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bh.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.bl.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dr.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dx.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.em.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.es.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.fo.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.fq.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oc.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oi.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qs.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qu.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |