Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.12.1.1 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}4&23\\13&0\end{bmatrix}$, $\begin{bmatrix}9&2\\24&23\end{bmatrix}$, $\begin{bmatrix}39&8\\12&23\end{bmatrix}$, $\begin{bmatrix}52&37\\51&32\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $258048$ |
Jacobian
| Conductor: | $2^{6}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 49x $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{7^4}\cdot\frac{67228x^{2}z^{2}-3920xy^{2}z+64y^{4}+117649z^{4}}{z^{2}x^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.6.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.24.1.d.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.e.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.j.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.k.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.dd.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.de.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.dh.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.di.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.7.e.1 | $56$ | $8$ | $8$ | $7$ | $4$ | $1^{6}$ |
| 56.252.19.e.1 | $56$ | $21$ | $21$ | $19$ | $6$ | $1^{2}\cdot2^{6}\cdot4$ |
| 56.336.25.e.1 | $56$ | $28$ | $28$ | $25$ | $9$ | $1^{8}\cdot2^{6}\cdot4$ |
| 168.24.1.cu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.cw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.cy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.da.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.mt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.mv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.mx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.mz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.3.a.1 | $168$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 168.48.3.by.1 | $168$ | $4$ | $4$ | $3$ | $?$ | not computed |
| 280.24.1.cu.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cy.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.da.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.mq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.ms.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.mu.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.mw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.60.5.a.1 | $280$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 280.72.5.a.1 | $280$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 280.120.9.mi.1 | $280$ | $10$ | $10$ | $9$ | $?$ | not computed |