Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.97 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&33\\31&32\end{bmatrix}$, $\begin{bmatrix}14&37\\51&30\end{bmatrix}$, $\begin{bmatrix}23&42\\50&29\end{bmatrix}$, $\begin{bmatrix}48&17\\51&4\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 112.48.1-56.bj.1.1, 112.48.1-56.bj.1.2, 112.48.1-56.bj.1.3, 112.48.1-56.bj.1.4, 112.48.1-56.bj.1.5, 112.48.1-56.bj.1.6, 112.48.1-56.bj.1.7, 112.48.1-56.bj.1.8 |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $129024$ |
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 $ | $=$ | $ 14 x y + 2 x z + z w $ |
| $=$ | $228 x^{2} + 4 x w - 14 y^{2} - 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 4 x^{3} z - 14 x^{2} y^{2} + 29 x^{2} z^{2} + 28 x z^{3} - 196 y^{2} z^{2} + 7 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{7}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{7}z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6\cdot3^3}{7^4}\cdot\frac{177187515984xz^{4}w+111072106992xz^{2}w^{3}+595035028xw^{5}+77112961632y^{2}z^{4}+84721610736y^{2}z^{2}w^{2}+14680131088y^{2}w^{4}-20344250304yz^{3}w^{2}-8314157712yzw^{4}+35328872232z^{6}+53473888116z^{4}w^{2}+14959337638z^{2}w^{4}+43378867w^{6}}{6238080xz^{4}w-4162368xz^{2}w^{3}-247828xw^{5}-21105504y^{2}z^{4}+4558176y^{2}z^{2}w^{2}+252938y^{2}w^{4}-1455552yz^{3}w^{2}-175560yzw^{4}-12060288z^{6}+1357056z^{4}w^{2}-500362z^{2}w^{4}-18067w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.12.0.bp.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.12.0.br.1 | $56$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.48.1.l.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ca.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.dm.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.do.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fm.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fo.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gb.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.gh.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.cp.1 | $56$ | $8$ | $8$ | $13$ | $6$ | $1^{12}$ |
| 56.504.37.et.1 | $56$ | $21$ | $21$ | $37$ | $9$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.et.1 | $56$ | $28$ | $28$ | $49$ | $15$ | $1^{20}\cdot2^{12}\cdot4$ |
| 168.48.1.pb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.pf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.qh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ql.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ve.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.wp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.fb.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.dv.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.of.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.oj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ui.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.uk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vt.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.cp.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.ed.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.bkf.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |