Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.731 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&20\\52&9\end{bmatrix}$, $\begin{bmatrix}19&6\\38&51\end{bmatrix}$, $\begin{bmatrix}51&14\\30&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.k.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $32256$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 7 y^{2} + z w $ |
| $=$ | $28 x^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 7 y^{2} z^{2} + 49 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.48.1.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+7Y^{2}Z^{2}+49Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.b.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.48.0-28.b.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-28.b.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.j.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.j.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1-8.b.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.768.25-56.bc.1.1 | $56$ | $8$ | $8$ | $25$ | $9$ | $1^{20}\cdot2^{2}$ |
| 56.2016.73-56.ck.1.1 | $56$ | $21$ | $21$ | $73$ | $25$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.2688.97-56.ck.1.1 | $56$ | $28$ | $28$ | $97$ | $34$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.192.5-112.i.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.i.2.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.j.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5-112.j.2.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.9-168.cs.1.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-168.bk.1.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.480.17-280.w.1.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |