Invariants
| Level: | $56$ | $\SL_2$-level: | $28$ | Newform level: | $3136$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $14^{6}\cdot28^{6}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28C16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.504.16.3 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&28\\14&27\end{bmatrix}$, $\begin{bmatrix}17&12\\12&25\end{bmatrix}$, $\begin{bmatrix}29&46\\52&15\end{bmatrix}$, $\begin{bmatrix}41&6\\36&13\end{bmatrix}$, $\begin{bmatrix}41&54\\40&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.252.16.b.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{64}\cdot7^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{6}$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.j, 3136.2.a.s |
Rational points
This modular curve has no $\Q_p$ points for $p=3,11,67$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.b.1.1 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.b.1.1 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 28.252.7-14.a.1.1 | $28$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
| 56.252.7-14.a.1.4 | $56$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1008.31-56.d.1.3 | $56$ | $2$ | $2$ | $31$ | $10$ | $1^{13}\cdot2$ |
| 56.1008.31-56.f.1.3 | $56$ | $2$ | $2$ | $31$ | $9$ | $1^{13}\cdot2$ |
| 56.1008.31-56.j.1.1 | $56$ | $2$ | $2$ | $31$ | $6$ | $1^{13}\cdot2$ |
| 56.1008.31-56.l.1.1 | $56$ | $2$ | $2$ | $31$ | $3$ | $1^{13}\cdot2$ |
| 56.1008.31-56.bb.1.1 | $56$ | $2$ | $2$ | $31$ | $4$ | $1^{13}\cdot2$ |
| 56.1008.31-56.bd.1.2 | $56$ | $2$ | $2$ | $31$ | $4$ | $1^{13}\cdot2$ |
| 56.1008.31-56.bh.1.2 | $56$ | $2$ | $2$ | $31$ | $7$ | $1^{13}\cdot2$ |
| 56.1008.31-56.bj.1.2 | $56$ | $2$ | $2$ | $31$ | $5$ | $1^{13}\cdot2$ |
| 56.1008.34-56.b.1.2 | $56$ | $2$ | $2$ | $34$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.b.1.8 | $56$ | $2$ | $2$ | $34$ | $4$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.c.1.4 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.c.1.20 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.i.1.2 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.i.1.6 | $56$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.j.1.10 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.j.1.12 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.bf.1.2 | $56$ | $2$ | $2$ | $34$ | $5$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bf.1.10 | $56$ | $2$ | $2$ | $34$ | $5$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bg.1.2 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bg.1.10 | $56$ | $2$ | $2$ | $34$ | $8$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bm.1.2 | $56$ | $2$ | $2$ | $34$ | $3$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bm.1.14 | $56$ | $2$ | $2$ | $34$ | $3$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bn.1.4 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.bn.1.10 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |