Invariants
| Level: | $112$ | $\SL_2$-level: | $16$ | Newform level: | $3136$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
Level structure
| $\GL_2(\Z/112\Z)$-generators: | $\begin{bmatrix}33&102\\5&51\end{bmatrix}$, $\begin{bmatrix}45&88\\91&79\end{bmatrix}$, $\begin{bmatrix}73&80\\69&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.da.1 for the level structure with $-I$) |
| Cyclic 112-isogeny field degree: | $32$ |
| Cyclic 112-torsion field degree: | $768$ |
| Full 112-torsion field degree: | $516096$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - x y + y^{2} - 4 z^{2} $ |
| $=$ | $11 x^{2} - 2 x y + 2 y^{2} + 8 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 450 x^{4} - 364 x^{2} y^{2} + 60 x^{2} z^{2} + 98 y^{4} - 21 y^{2} z^{2} + 2 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(4z^{2}+w^{2})^{3}(12z^{2}+w^{2})^{3}}{z^{8}(8z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.48.1.da.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 450X^{4}-364X^{2}Y^{2}+98Y^{4}+60X^{2}Z^{2}-21Y^{2}Z^{2}+2Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-8.s.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 112.48.0-8.s.1.4 | $112$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |