Invariants
Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16J3 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}13&24\\0&9\end{bmatrix}$, $\begin{bmatrix}73&16\\48&77\end{bmatrix}$, $\begin{bmatrix}75&34\\32&45\end{bmatrix}$, $\begin{bmatrix}111&12\\40&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.96.3.ce.2 for the level structure with $-I$) |
Cyclic 112-isogeny field degree: | $16$ |
Cyclic 112-torsion field degree: | $768$ |
Full 112-torsion field degree: | $258048$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.96.1-16.a.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ |
56.96.0-56.bb.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
112.96.0-56.bb.1.5 | $112$ | $2$ | $2$ | $0$ | $?$ |
112.96.1-16.a.2.12 | $112$ | $2$ | $2$ | $1$ | $?$ |
112.96.2-112.d.2.11 | $112$ | $2$ | $2$ | $2$ | $?$ |
112.96.2-112.d.2.19 | $112$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
112.384.5-112.bg.3.3 | $112$ | $2$ | $2$ | $5$ |
112.384.5-112.bi.2.4 | $112$ | $2$ | $2$ | $5$ |
112.384.5-112.ed.2.5 | $112$ | $2$ | $2$ | $5$ |
112.384.5-112.ee.2.4 | $112$ | $2$ | $2$ | $5$ |
224.384.9-224.q.1.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.q.2.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.r.2.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.r.3.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.bp.1.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.bp.2.11 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.bq.1.10 | $224$ | $2$ | $2$ | $9$ |
224.384.9-224.bq.3.10 | $224$ | $2$ | $2$ | $9$ |