Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
Genus: | $49 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
Cusps: | $32$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot28^{8}\cdot56^{8}$ | Cusp orbits | $2^{14}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $3$ | ||||||
$\Q$-gonality: | $8 \le \gamma \le 16$ | ||||||
$\overline{\Q}$-gonality: | $8 \le \gamma \le 16$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.49.846 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}31&32\\12&15\end{bmatrix}$, $\begin{bmatrix}33&4\\0&25\end{bmatrix}$, $\begin{bmatrix}33&36\\44&25\end{bmatrix}$, $\begin{bmatrix}35&36\\24&17\end{bmatrix}$, $\begin{bmatrix}39&20\\24&39\end{bmatrix}$, $\begin{bmatrix}53&0\\12&17\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.1536.49-56.fu.4.1, 56.1536.49-56.fu.4.2, 56.1536.49-56.fu.4.3, 56.1536.49-56.fu.4.4, 56.1536.49-56.fu.4.5, 56.1536.49-56.fu.4.6, 56.1536.49-56.fu.4.7, 56.1536.49-56.fu.4.8, 56.1536.49-56.fu.4.9, 56.1536.49-56.fu.4.10, 56.1536.49-56.fu.4.11, 56.1536.49-56.fu.4.12, 56.1536.49-56.fu.4.13, 56.1536.49-56.fu.4.14, 56.1536.49-56.fu.4.15, 56.1536.49-56.fu.4.16, 56.1536.49-56.fu.4.17, 56.1536.49-56.fu.4.18, 56.1536.49-56.fu.4.19, 56.1536.49-56.fu.4.20, 56.1536.49-56.fu.4.21, 56.1536.49-56.fu.4.22, 56.1536.49-56.fu.4.23, 56.1536.49-56.fu.4.24 |
Cyclic 56-isogeny field degree: | $2$ |
Cyclic 56-torsion field degree: | $24$ |
Full 56-torsion field degree: | $4032$ |
Jacobian
Conductor: | $2^{196}\cdot7^{49}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{11}\cdot2^{7}\cdot4^{4}\cdot8$ |
Newforms: | 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 56.2.e.a, 56.2.e.b, 112.2.a.a, 112.2.a.b, 112.2.a.c, 224.2.e.a, 224.2.e.b, 448.2.f.a, 448.2.f.b, 448.2.f.c, 448.2.f.d |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.384.23.e.4 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
56.384.23.i.1 | $56$ | $2$ | $2$ | $23$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
56.384.23.bn.4 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2^{2}\cdot4^{2}\cdot8$ |
56.384.23.bv.3 | $56$ | $2$ | $2$ | $23$ | $2$ | $1^{6}\cdot2^{2}\cdot4^{2}\cdot8$ |
56.384.25.bm.3 | $56$ | $2$ | $2$ | $25$ | $3$ | $2^{4}\cdot4^{4}$ |
56.384.25.fd.4 | $56$ | $2$ | $2$ | $25$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
56.384.25.fl.4 | $56$ | $2$ | $2$ | $25$ | $0$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.1536.97.bq.4 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.br.4 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.bw.4 | $56$ | $2$ | $2$ | $97$ | $11$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.bx.4 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.fu.3 | $56$ | $2$ | $2$ | $97$ | $8$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.fv.3 | $56$ | $2$ | $2$ | $97$ | $10$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.ga.3 | $56$ | $2$ | $2$ | $97$ | $11$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.97.gb.3 | $56$ | $2$ | $2$ | $97$ | $7$ | $1^{10}\cdot2^{11}\cdot4^{4}$ |
56.1536.105.gb.2 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{11}\cdot4^{4}\cdot8$ |
56.1536.105.gf.4 | $56$ | $2$ | $2$ | $105$ | $5$ | $1^{10}\cdot2^{11}\cdot4^{4}\cdot8$ |
56.1536.105.gi.2 | $56$ | $2$ | $2$ | $105$ | $12$ | $1^{10}\cdot2^{11}\cdot4^{4}\cdot8$ |
56.1536.105.gk.4 | $56$ | $2$ | $2$ | $105$ | $9$ | $1^{10}\cdot2^{11}\cdot4^{4}\cdot8$ |
56.1536.105.go.2 | $56$ | $2$ | $2$ | $105$ | $10$ | $1^{10}\cdot2^{7}\cdot4^{2}\cdot8^{3}$ |
56.1536.105.gp.2 | $56$ | $2$ | $2$ | $105$ | $12$ | $1^{10}\cdot2^{7}\cdot4^{2}\cdot8^{3}$ |
56.1536.105.gq.6 | $56$ | $2$ | $2$ | $105$ | $9$ | $1^{10}\cdot2^{7}\cdot4^{2}\cdot8^{3}$ |
56.1536.105.gr.6 | $56$ | $2$ | $2$ | $105$ | $5$ | $1^{10}\cdot2^{7}\cdot4^{2}\cdot8^{3}$ |
56.2304.145.nb.1 | $56$ | $3$ | $3$ | $145$ | $3$ | $2^{12}\cdot4^{2}\cdot12^{4}\cdot16$ |
56.2304.145.nc.2 | $56$ | $3$ | $3$ | $145$ | $5$ | $2^{12}\cdot4^{2}\cdot12^{4}\cdot16$ |
56.2304.145.tt.4 | $56$ | $3$ | $3$ | $145$ | $11$ | $1^{20}\cdot2^{2}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16$ |
56.5376.385.jo.1 | $56$ | $7$ | $7$ | $385$ | $22$ | $1^{48}\cdot2^{28}\cdot4^{16}\cdot6^{4}\cdot8^{8}\cdot12^{4}\cdot16^{2}$ |