Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| 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 | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.1.627 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&40\\14&53\end{bmatrix}$, $\begin{bmatrix}9&30\\26&53\end{bmatrix}$, $\begin{bmatrix}29&30\\52&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.1.ce.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{6}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y^{2} + z^{2} $ |
| $=$ | $14 y^{2} - 14 z^{2} + w^{2}$ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7^4}\cdot\frac{(38416z^{8}-5488z^{6}w^{2}+980z^{4}w^{4}-56z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(14z^{2}-w^{2})^{4}(28z^{2}-w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.j.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.j.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.k.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.k.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.l.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.l.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.ba.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.ba.1.10 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.1-56.bg.2.9 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bg.2.11 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bh.1.9 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bh.1.11 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bv.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.bv.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.49-56.of.1.14 | $56$ | $8$ | $8$ | $49$ | $8$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.blc.2.8 | $56$ | $21$ | $21$ | $145$ | $24$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.bme.1.12 | $56$ | $28$ | $28$ | $193$ | $31$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 112.384.5-112.x.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cf.1.5 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.ea.1.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.el.1.5 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |