Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.140 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&36\\48&29\end{bmatrix}$, $\begin{bmatrix}14&55\\47&14\end{bmatrix}$, $\begin{bmatrix}24&1\\9&32\end{bmatrix}$, $\begin{bmatrix}43&48\\4&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.1.n.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $64512$ |
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 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{7^2}\cdot\frac{1735923x^{2}y^{4}z^{2}-1156736144655x^{2}z^{6}-2254xy^{6}z+47231014593xy^{2}z^{5}+y^{8}-487066860y^{4}z^{4}+13841287201z^{8}}{zy^{4}(49x^{2}z+xy^{2}+2401z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-4.d.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0-4.d.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.96.1-56.cv.1.4 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.cy.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dg.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dn.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dz.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.ea.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.en.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.eo.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.384.13-56.bf.1.19 | $56$ | $8$ | $8$ | $13$ | $5$ | $1^{8}\cdot2^{2}$ |
| 56.1008.37-56.bz.1.15 | $56$ | $21$ | $21$ | $37$ | $13$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.1344.49-56.bz.1.11 | $56$ | $28$ | $28$ | $49$ | $17$ | $1^{12}\cdot2^{16}\cdot4$ |
| 168.96.1-168.ji.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.jm.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.lc.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ln.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ml.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mq.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.oc.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.og.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.5-168.bx.1.28 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.192.5-168.bs.1.4 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.96.1-280.iw.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ja.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ke.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.kp.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ln.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ls.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ne.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.ni.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.240.9-280.z.1.6 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.288.9-280.bt.1.15 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.480.17-280.tl.1.24 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |