Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $1568$ | ||
| 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: | 56.48.1.139 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&22\\50&55\end{bmatrix}$, $\begin{bmatrix}9&32\\20&37\end{bmatrix}$, $\begin{bmatrix}12&1\\21&52\end{bmatrix}$, $\begin{bmatrix}22&13\\41&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.1.m.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: | $2^{5}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1568.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 196x $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{7^2}\cdot\frac{27774768x^{2}y^{4}z^{2}-1184497812126720x^{2}z^{6}-9016xy^{6}z+12091139735808xy^{2}z^{5}+y^{8}-31172279040y^{4}z^{4}+56693912375296z^{8}}{zy^{4}(196x^{2}z+xy^{2}+38416z^{3})}$ |
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 |
| 28.24.0-4.d.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ | 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.cw.1.4 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.cx.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dj.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dk.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.dw.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.ed.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.em.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.96.1-56.ep.1.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.384.13-56.be.1.19 | $56$ | $8$ | $8$ | $13$ | $5$ | $1^{8}\cdot2^{2}$ |
| 56.1008.37-56.by.1.15 | $56$ | $21$ | $21$ | $37$ | $13$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.1344.49-56.by.1.10 | $56$ | $28$ | $28$ | $49$ | $17$ | $1^{12}\cdot2^{16}\cdot4$ |
| 168.96.1-168.jh.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.jl.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.lf.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.lk.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mi.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mt.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ob.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.of.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.5-168.bw.1.15 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.192.5-168.br.1.4 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.96.1-280.iv.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.iz.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.kh.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.km.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.lk.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.lv.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.nd.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.nh.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.240.9-280.y.1.11 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.288.9-280.bs.1.12 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.480.17-280.tk.1.28 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |