Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $10752$ | $\PSL_2$-index: | $5376$ | ||||
Genus: | $401 = 1 + \frac{ 5376 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 96 }{2}$ | ||||||
Cusps: | $96$ (none of which are rational) | Cusp widths | $56^{96}$ | Cusp orbits | $2^{2}\cdot4^{5}\cdot6^{2}\cdot12^{3}\cdot24$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $79$ | ||||||
$\Q$-gonality: | $54 \le \gamma \le 112$ | ||||||
$\overline{\Q}$-gonality: | $54 \le \gamma \le 112$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.10752.401.9471 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&4\\50&25\end{bmatrix}$, $\begin{bmatrix}13&24\\34&23\end{bmatrix}$, $\begin{bmatrix}25&48\\40&17\end{bmatrix}$, $\begin{bmatrix}35&8\\54&21\end{bmatrix}$ |
$\GL_2(\Z/56\Z)$-subgroup: | $C_6^2:D_4$ |
Contains $-I$: | no $\quad$ (see 56.5376.401.lb.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $4$ |
Cyclic 56-torsion field degree: | $48$ |
Full 56-torsion field degree: | $288$ |
Jacobian
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=5,29,31,53,79,149,223,389$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.5376.193-56.lc.2.8 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
56.5376.193-56.lc.2.19 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
56.5376.193-56.lh.1.12 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{2}\cdot8\cdot12^{3}$ |
56.5376.193-56.lh.1.17 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{2}\cdot8\cdot12^{3}$ |
56.5376.193-56.ob.1.4 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{2}\cdot8\cdot12^{3}$ |
56.5376.193-56.ob.1.18 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{2}\cdot8\cdot12^{3}$ |
56.5376.201-56.bg.1.6 | $56$ | $2$ | $2$ | $201$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
56.5376.201-56.bg.1.22 | $56$ | $2$ | $2$ | $201$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
56.5376.201-56.di.1.11 | $56$ | $2$ | $2$ | $201$ | $29$ | $1^{64}\cdot2^{24}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.di.1.29 | $56$ | $2$ | $2$ | $201$ | $29$ | $1^{64}\cdot2^{24}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.dj.1.7 | $56$ | $2$ | $2$ | $201$ | $79$ | $2^{16}\cdot4^{16}\cdot6^{6}\cdot8\cdot12^{5}$ |
56.5376.201-56.dj.1.14 | $56$ | $2$ | $2$ | $201$ | $79$ | $2^{16}\cdot4^{16}\cdot6^{6}\cdot8\cdot12^{5}$ |
56.5376.201-56.dz.2.6 | $56$ | $2$ | $2$ | $201$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
56.5376.201-56.dz.2.31 | $56$ | $2$ | $2$ | $201$ | $32$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.21504.801-56.fg.1.9 | $56$ | $2$ | $2$ | $801$ | $159$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.gd.1.8 | $56$ | $2$ | $2$ | $801$ | $153$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.kw.2.8 | $56$ | $2$ | $2$ | $801$ | $161$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.lo.1.8 | $56$ | $2$ | $2$ | $801$ | $157$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.uo.2.8 | $56$ | $2$ | $2$ | $801$ | $150$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.uv.2.6 | $56$ | $2$ | $2$ | $801$ | $169$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.wx.3.8 | $56$ | $2$ | $2$ | $801$ | $138$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.21504.801-56.xc.1.6 | $56$ | $2$ | $2$ | $801$ | $165$ | $1^{102}\cdot2^{63}\cdot4^{21}\cdot6^{6}\cdot12^{3}\cdot16$ |
56.32256.1201-56.no.2.6 | $56$ | $3$ | $3$ | $1201$ | $247$ | $1^{190}\cdot2^{123}\cdot4^{31}\cdot6^{18}\cdot8\cdot12^{9}\cdot16$ |