Invariants
Level: | $112$ | $\SL_2$-level: | $16$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4\cdot16$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16C0 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}10&49\\99&88\end{bmatrix}$, $\begin{bmatrix}40&59\\105&82\end{bmatrix}$, $\begin{bmatrix}82&13\\87&64\end{bmatrix}$, $\begin{bmatrix}88&71\\89&54\end{bmatrix}$, $\begin{bmatrix}92&77\\93&108\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.24.0.h.1 for the level structure with $-I$) |
Cyclic 112-isogeny field degree: | $16$ |
Cyclic 112-torsion field degree: | $768$ |
Full 112-torsion field degree: | $1032192$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.n.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
112.24.0-8.n.1.3 | $112$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
112.96.0-112.bk.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bk.2.11 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bl.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bl.2.10 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bm.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bm.2.13 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bn.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bn.2.11 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bo.1.11 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bo.2.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bp.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bp.2.11 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bq.1.10 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bq.2.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.br.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.br.2.10 | $112$ | $2$ | $2$ | $0$ |
112.96.1-112.a.2.4 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.f.1.14 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.g.1.16 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.j.1.14 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.q.1.6 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.t.1.10 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.u.1.14 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.x.1.10 | $112$ | $2$ | $2$ | $1$ |
112.384.11-112.v.1.53 | $112$ | $8$ | $8$ | $11$ |