Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $384$ | $\PSL_2$-index: | $384$ | ||||
Genus: | $25 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot28^{4}\cdot56^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $13$ | ||||||
$\Q$-gonality: | $8$ | ||||||
$\overline{\Q}$-gonality: | $8$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | yes $\quad(D =$ $-7$) |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.384.25.175 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&23\\36&49\end{bmatrix}$, $\begin{bmatrix}21&16\\12&9\end{bmatrix}$, $\begin{bmatrix}37&28\\54&43\end{bmatrix}$, $\begin{bmatrix}53&4\\46&19\end{bmatrix}$, $\begin{bmatrix}55&23\\22&41\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.768.25-56.qy.1.1, 56.768.25-56.qy.1.2, 56.768.25-56.qy.1.3, 56.768.25-56.qy.1.4, 56.768.25-56.qy.1.5, 56.768.25-56.qy.1.6, 56.768.25-56.qy.1.7, 56.768.25-56.qy.1.8, 56.768.25-56.qy.1.9, 56.768.25-56.qy.1.10 |
Cyclic 56-isogeny field degree: | $4$ |
Cyclic 56-torsion field degree: | $96$ |
Full 56-torsion field degree: | $8064$ |
Jacobian
Conductor: | $2^{112}\cdot7^{37}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{21}\cdot2^{2}$ |
Newforms: | 14.2.a.a$^{2}$, 56.2.a.a, 56.2.a.b, 98.2.a.a$^{2}$, 112.2.a.a, 112.2.a.b, 392.2.a.b, 392.2.a.d, 448.2.a.c, 448.2.a.d, 448.2.a.g, 448.2.a.i, 448.2.a.j, 784.2.a.b, 3136.2.a.c, 3136.2.a.f, 3136.2.a.m$^{2}$, 3136.2.a.p, 3136.2.a.y, 3136.2.a.z |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
28.192.11.x.1 | $28$ | $2$ | $2$ | $11$ | $3$ | $1^{10}\cdot2^{2}$ |
56.48.1.jf.1 | $56$ | $8$ | $8$ | $1$ | $1$ | $1^{20}\cdot2^{2}$ |
56.192.11.gr.1 | $56$ | $2$ | $2$ | $11$ | $6$ | $1^{10}\cdot2^{2}$ |
56.192.13.dt.1 | $56$ | $2$ | $2$ | $13$ | $8$ | $1^{8}\cdot2^{2}$ |
56.192.13.el.1 | $56$ | $2$ | $2$ | $13$ | $7$ | $1^{8}\cdot2^{2}$ |
56.192.13.fa.1 | $56$ | $2$ | $2$ | $13$ | $8$ | $1^{12}$ |
56.192.13.hd.1 | $56$ | $2$ | $2$ | $13$ | $7$ | $1^{12}$ |
56.192.13.hv.1 | $56$ | $2$ | $2$ | $13$ | $4$ | $1^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.768.53.fr.1 | $56$ | $2$ | $2$ | $53$ | $25$ | $1^{20}\cdot2^{4}$ |
56.768.53.fs.1 | $56$ | $2$ | $2$ | $53$ | $21$ | $1^{20}\cdot2^{4}$ |
56.1152.73.blq.1 | $56$ | $3$ | $3$ | $73$ | $15$ | $2^{16}\cdot4^{4}$ |
56.1152.73.blq.2 | $56$ | $3$ | $3$ | $73$ | $15$ | $2^{16}\cdot4^{4}$ |
56.1152.73.bok.1 | $56$ | $3$ | $3$ | $73$ | $55$ | $1^{32}\cdot2^{8}$ |
56.2688.193.eip.1 | $56$ | $7$ | $7$ | $193$ | $72$ | $1^{86}\cdot2^{39}\cdot4$ |