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 | $2^{2}\cdot4^{3}\cdot6^{2}\cdot8\cdot12^{3}\cdot24$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $67$ | ||||||
$\Q$-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: | 56.10752.401.284 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}21&24\\20&31\end{bmatrix}$, $\begin{bmatrix}27&28\\0&13\end{bmatrix}$, $\begin{bmatrix}47&16\\36&51\end{bmatrix}$, $\begin{bmatrix}49&20\\40&41\end{bmatrix}$ |
$\GL_2(\Z/56\Z)$-subgroup: | $C_6^2:D_4$ |
Contains $-I$: | no $\quad$ (see 56.5376.401.ko.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 $p=5,31,79,173,223,269,293,941$, 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 |
---|---|---|---|---|---|---|
56.384.5-56.e.2.7 | $56$ | $28$ | $28$ | $5$ | $2$ | $1^{66}\cdot2^{69}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$ |
56.5376.193-56.lc.2.8 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.193-56.lc.2.22 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.193-56.li.1.9 | $56$ | $2$ | $2$ | $193$ | $26$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.193-56.li.1.24 | $56$ | $2$ | $2$ | $193$ | $26$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.193-56.oc.1.3 | $56$ | $2$ | $2$ | $193$ | $26$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.193-56.oc.1.30 | $56$ | $2$ | $2$ | $193$ | $26$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
56.5376.201-56.bc.1.6 | $56$ | $2$ | $2$ | $201$ | $26$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.bc.1.23 | $56$ | $2$ | $2$ | $201$ | $26$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.cq.2.5 | $56$ | $2$ | $2$ | $201$ | $29$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.cq.2.32 | $56$ | $2$ | $2$ | $201$ | $29$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.cr.2.7 | $56$ | $2$ | $2$ | $201$ | $67$ | $2^{10}\cdot4^{11}\cdot6^{6}\cdot12^{7}\cdot16$ |
56.5376.201-56.cr.2.22 | $56$ | $2$ | $2$ | $201$ | $67$ | $2^{10}\cdot4^{11}\cdot6^{6}\cdot12^{7}\cdot16$ |
56.5376.201-56.cz.2.6 | $56$ | $2$ | $2$ | $201$ | $26$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
56.5376.201-56.cz.2.31 | $56$ | $2$ | $2$ | $201$ | $26$ | $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$ |
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.fr.1.9 | $56$ | $2$ | $2$ | $801$ | $137$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.gg.1.8 | $56$ | $2$ | $2$ | $801$ | $161$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.kz.1.8 | $56$ | $2$ | $2$ | $801$ | $135$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.lo.1.8 | $56$ | $2$ | $2$ | $801$ | $157$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.uo.2.8 | $56$ | $2$ | $2$ | $801$ | $150$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.uw.1.6 | $56$ | $2$ | $2$ | $801$ | $143$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.xa.2.8 | $56$ | $2$ | $2$ | $801$ | $146$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.21504.801-56.xi.1.6 | $56$ | $2$ | $2$ | $801$ | $143$ | $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$ |
56.32256.1201-56.ni.2.8 | $56$ | $3$ | $3$ | $1201$ | $211$ | $1^{190}\cdot2^{123}\cdot4^{31}\cdot6^{18}\cdot8\cdot12^{9}\cdot16$ |