Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $23 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot28^{8}\cdot56^{2}$ | Cusp orbits | $2^{10}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 56P23 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.768.23.15 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&8\\48&25\end{bmatrix}$, $\begin{bmatrix}13&12\\24&15\end{bmatrix}$, $\begin{bmatrix}27&44\\18&39\end{bmatrix}$, $\begin{bmatrix}51&52\\14&47\end{bmatrix}$, $\begin{bmatrix}55&20\\42&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.384.23.n.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $24$ |
| Full 56-torsion field degree: | $4032$ |
Jacobian
| Conductor: | $2^{93}\cdot7^{23}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{2}\cdot4^{2}$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 224.2.b.a, 224.2.b.b, 448.2.a.a, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.g, 448.2.a.h |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.e.2.6 | $8$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.e.2.6 | $8$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 56.384.11-56.c.1.14 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{2}\cdot4^{2}$ |
| 56.384.11-56.c.1.15 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{2}\cdot4^{2}$ |
| 56.384.11-56.p.1.8 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.p.1.58 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.s.1.4 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.384.11-56.s.1.53 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1536.45-56.j.1.8 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.j.3.6 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.l.1.8 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.l.3.7 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bh.1.8 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bh.2.6 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bj.1.8 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.1536.45-56.bj.2.7 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.1536.49-56.dn.1.5 | $56$ | $2$ | $2$ | $49$ | $5$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.gb.1.5 | $56$ | $2$ | $2$ | $49$ | $5$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.gh.1.7 | $56$ | $2$ | $2$ | $49$ | $9$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.gn.2.7 | $56$ | $2$ | $2$ | $49$ | $9$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.gv.1.7 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.gv.3.5 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.gv.5.5 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.gv.7.1 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.hi.1.4 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.hi.3.3 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.hi.5.3 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.hi.7.1 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.1536.49-56.ho.1.15 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.hq.1.15 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.ht.1.16 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1536.49-56.hw.1.16 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.2304.67-56.t.2.19 | $56$ | $3$ | $3$ | $67$ | $4$ | $2^{10}\cdot12^{2}$ |
| 56.2304.67-56.t.4.20 | $56$ | $3$ | $3$ | $67$ | $4$ | $2^{10}\cdot12^{2}$ |
| 56.2304.67-56.bh.1.16 | $56$ | $3$ | $3$ | $67$ | $10$ | $1^{20}\cdot6^{4}$ |
| 56.5376.185-56.v.1.19 | $56$ | $7$ | $7$ | $185$ | $27$ | $1^{48}\cdot2^{21}\cdot4^{6}\cdot6^{4}\cdot12^{2}$ |