Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}7&40\\48&13\end{bmatrix}$, $\begin{bmatrix}9&28\\80&37\end{bmatrix}$, $\begin{bmatrix}87&36\\16&65\end{bmatrix}$, $\begin{bmatrix}87&68\\84&99\end{bmatrix}$, $\begin{bmatrix}105&20\\92&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.f.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $32$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $258048$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 x w - y^{2} $ |
| $=$ | $y^{2} + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{8}-2z^{6}w^{2}+5z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(z-w)^{4}(z+w)^{4}(2z^{2}-w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.1.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-2X^{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 |
|---|---|---|---|---|---|---|
| 112.96.0-8.c.1.2 | $112$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 112.96.0-8.c.1.3 | $112$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 112.384.5-16.b.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.b.1.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-8.c.1.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-8.c.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-8.d.3.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-8.d.3.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.i.1.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.i.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.j.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.j.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.o.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.o.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.v.1.5 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.v.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-56.z.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-56.z.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-56.ba.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-56.ba.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bl.1.13 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bl.1.14 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bs.1.13 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bs.1.15 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cx.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cx.2.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |