Invariants
Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $784$ | ||
Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot6$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $5$ | ||||||
$\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
$\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 28D21 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.691 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&47\\35&10\end{bmatrix}$, $\begin{bmatrix}6&1\\53&50\end{bmatrix}$, $\begin{bmatrix}9&42\\34&33\end{bmatrix}$, $\begin{bmatrix}10&47\\23&46\end{bmatrix}$, $\begin{bmatrix}32&17\\33&52\end{bmatrix}$, $\begin{bmatrix}47&36\\0&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 28.336.21.p.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $4$ |
Cyclic 56-torsion field degree: | $96$ |
Full 56-torsion field degree: | $4608$ |
Jacobian
Conductor: | $2^{60}\cdot7^{37}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{9}\cdot2^{6}$ |
Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.24.0-4.d.1.4 | $56$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
56.336.9-28.a.1.6 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
56.336.9-28.a.1.8 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.1344.41-28.ba.1.1 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{18}\cdot2$ |
56.1344.41-28.bb.1.2 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{18}\cdot2$ |
56.1344.41-28.be.1.1 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{18}\cdot2$ |
56.1344.41-28.bf.1.2 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
56.1344.41-56.hc.1.2 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{18}\cdot2$ |
56.1344.41-56.hj.1.5 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
56.1344.41-56.ie.1.3 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{18}\cdot2$ |
56.1344.41-56.il.1.3 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{18}\cdot2$ |
56.1344.45-28.t.1.3 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{12}\cdot2^{6}$ |
56.1344.45-28.u.1.3 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{12}\cdot2^{6}$ |
56.1344.45-28.v.1.1 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
56.1344.45-28.w.1.2 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.di.1.14 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.dj.1.19 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.dm.1.14 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.dn.1.14 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.ds.1.12 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.dt.1.14 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.dw.1.15 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.dx.1.15 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.ea.1.4 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.eb.1.4 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.ec.1.1 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.ed.1.5 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.em.1.14 | $56$ | $2$ | $2$ | $45$ | $9$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.en.1.14 | $56$ | $2$ | $2$ | $45$ | $19$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.eq.1.12 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.er.1.14 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{20}\cdot2^{2}$ |
56.1344.45-56.eu.1.14 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.ev.1.14 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.ey.1.14 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{12}\cdot2^{6}$ |
56.1344.45-56.ez.1.11 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{12}\cdot2^{6}$ |
56.1344.49-56.bw.1.16 | $56$ | $2$ | $2$ | $49$ | $14$ | $1^{4}\cdot2^{10}\cdot4$ |
56.1344.49-56.bx.1.14 | $56$ | $2$ | $2$ | $49$ | $16$ | $1^{4}\cdot2^{10}\cdot4$ |
56.1344.49-56.by.1.14 | $56$ | $2$ | $2$ | $49$ | $17$ | $1^{4}\cdot2^{10}\cdot4$ |
56.1344.49-56.bz.1.11 | $56$ | $2$ | $2$ | $49$ | $17$ | $1^{4}\cdot2^{10}\cdot4$ |
56.1344.49-56.ca.1.11 | $56$ | $2$ | $2$ | $49$ | $19$ | $1^{12}\cdot2^{8}$ |
56.1344.49-56.cb.1.14 | $56$ | $2$ | $2$ | $49$ | $15$ | $1^{12}\cdot2^{8}$ |
56.1344.49-56.cc.1.14 | $56$ | $2$ | $2$ | $49$ | $19$ | $1^{12}\cdot2^{8}$ |
56.1344.49-56.cd.1.16 | $56$ | $2$ | $2$ | $49$ | $15$ | $1^{12}\cdot2^{8}$ |
56.2016.61-28.t.1.3 | $56$ | $3$ | $3$ | $61$ | $16$ | $1^{26}\cdot2^{7}$ |