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 | $1^{2}\cdot2^{3}\cdot16$ | 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: | 16D0 |
Level structure
$\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}27&88\\30&45\end{bmatrix}$, $\begin{bmatrix}69&6\\10&41\end{bmatrix}$, $\begin{bmatrix}78&103\\11&10\end{bmatrix}$, $\begin{bmatrix}86&99\\99&70\end{bmatrix}$, $\begin{bmatrix}109&64\\48&109\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 112.24.0.f.2 for the level structure with $-I$) |
Cyclic 112-isogeny field degree: | $16$ |
Cyclic 112-torsion field degree: | $384$ |
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 |
---|---|---|---|---|---|
16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
56.24.0-8.n.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ |
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.d.1.21 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.f.1.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.k.1.6 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.l.2.10 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.u.1.13 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.x.2.9 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.z.1.2 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.ba.2.10 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bg.2.3 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bh.1.1 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bo.1.1 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bp.2.3 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bu.2.3 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bv.1.1 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.by.1.1 | $112$ | $2$ | $2$ | $0$ |
112.96.0-112.bz.2.3 | $112$ | $2$ | $2$ | $0$ |
112.96.1-112.bi.1.5 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.bj.1.1 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.bm.1.1 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.bn.1.5 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.bs.1.5 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.bt.1.1 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.ca.1.1 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.cb.1.5 | $112$ | $2$ | $2$ | $1$ |
112.384.11-112.r.1.6 | $112$ | $8$ | $8$ | $11$ |