Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | 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 | $2^{4}$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.6 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&51\\44&9\end{bmatrix}$, $\begin{bmatrix}29&8\\0&33\end{bmatrix}$, $\begin{bmatrix}43&39\\0&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.dg.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $32256$ |
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 $ | $=$ | $ 2 x^{2} + 2 x y - 4 x z - y^{2} $ |
| $=$ | $x^{2} + x y - 2 x z + 3 y^{2} - 7 y z + 7 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 14 x^{2} y^{2} + 4 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot7^2\,\frac{z^{3}(7z^{2}-4w^{2})(588yz^{4}w^{2}-336yz^{2}w^{4}-64yw^{6}-343z^{7}-196z^{5}w^{2}+560z^{3}w^{4}-128zw^{6})}{w^{8}(28yzw^{2}-49z^{4}+4w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.48.1.dg.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{7}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-14X^{2}Y^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.48.0-4.c.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-4.c.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.t.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.t.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1-56.n.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1-56.n.1.8 | $56$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 56.768.25-56.ja.1.5 | $56$ | $8$ | $8$ | $25$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.2016.73-56.yy.1.1 | $56$ | $21$ | $21$ | $73$ | $24$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.2688.97-56.yi.1.1 | $56$ | $28$ | $28$ | $97$ | $35$ | $1^{36}\cdot2^{28}\cdot4$ |
| 168.288.9-168.csm.1.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-168.zz.1.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.480.17-280.om.1.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |