Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
Genus: | $31 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $7^{12}\cdot14^{6}\cdot56^{6}$ | Cusp orbits | $6^{2}\cdot12$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $24$ | ||||||
$\Q$-gonality: | $10 \le \gamma \le 16$ | ||||||
$\overline{\Q}$-gonality: | $10 \le \gamma \le 16$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.31.2721 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&6\\8&23\end{bmatrix}$, $\begin{bmatrix}13&20\\38&43\end{bmatrix}$, $\begin{bmatrix}14&17\\41&42\end{bmatrix}$, $\begin{bmatrix}26&29\\47&28\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.504.31.oz.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $192$ |
Full 56-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{142}\cdot7^{62}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{17}\cdot2^{7}$ |
Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 784.2.a.b$^{2}$, 784.2.a.f$^{3}$, 784.2.a.g, 3136.2.a.a, 3136.2.a.b, 3136.2.a.bk, 3136.2.a.bn, 3136.2.a.bq, 3136.2.a.br, 3136.2.a.c, 3136.2.a.h, 3136.2.a.k, 3136.2.a.o, 3136.2.a.p, 3136.2.a.t, 3136.2.a.u, 3136.2.a.z |
Rational points
This modular curve has 2 rational CM points 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 |
---|---|---|---|---|---|---|
56.504.13-28.j.1.8 | $56$ | $2$ | $2$ | $13$ | $6$ | $1^{10}\cdot2^{4}$ |
56.504.13-28.j.1.9 | $56$ | $2$ | $2$ | $13$ | $6$ | $1^{10}\cdot2^{4}$ |
56.504.16-56.ct.1.15 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{9}\cdot2^{3}$ |
56.504.16-56.ct.1.27 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{9}\cdot2^{3}$ |
56.504.16-56.cx.1.7 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{13}\cdot2$ |
56.504.16-56.cx.1.11 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{13}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.2016.67-56.ht.1.2 | $56$ | $2$ | $2$ | $67$ | $28$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.it.1.7 | $56$ | $2$ | $2$ | $67$ | $28$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.uj.1.1 | $56$ | $2$ | $2$ | $67$ | $49$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.uk.1.3 | $56$ | $2$ | $2$ | $67$ | $49$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.wm.1.2 | $56$ | $2$ | $2$ | $67$ | $37$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.wq.1.1 | $56$ | $2$ | $2$ | $67$ | $37$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.xf.1.2 | $56$ | $2$ | $2$ | $67$ | $38$ | $1^{20}\cdot2^{8}$ |
56.2016.67-56.xj.1.1 | $56$ | $2$ | $2$ | $67$ | $38$ | $1^{20}\cdot2^{8}$ |