Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $1344$ | $\PSL_2$-index: | $1344$ | ||||
Genus: | $93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
Cusps: | $40$ (of which $2$ are rational) | Cusp widths | $14^{16}\cdot28^{8}\cdot56^{16}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot3^{2}\cdot4\cdot6^{2}\cdot12$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $12$ | ||||||
$\Q$-gonality: | $14 \le \gamma \le 28$ | ||||||
$\overline{\Q}$-gonality: | $14 \le \gamma \le 28$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1344.93.25 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&14\\0&51\end{bmatrix}$, $\begin{bmatrix}22&19\\47&34\end{bmatrix}$, $\begin{bmatrix}38&49\\53&2\end{bmatrix}$, $\begin{bmatrix}45&0\\10&39\end{bmatrix}$, $\begin{bmatrix}47&4\\10&37\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.2688.93-56.ft.1.1, 56.2688.93-56.ft.1.2, 56.2688.93-56.ft.1.3, 56.2688.93-56.ft.1.4, 56.2688.93-56.ft.1.5, 56.2688.93-56.ft.1.6, 56.2688.93-56.ft.1.7, 56.2688.93-56.ft.1.8, 56.2688.93-56.ft.1.9, 56.2688.93-56.ft.1.10, 56.2688.93-56.ft.1.11, 56.2688.93-56.ft.1.12, 56.2688.93-56.ft.1.13, 56.2688.93-56.ft.1.14, 56.2688.93-56.ft.1.15, 56.2688.93-56.ft.1.16 |
Cyclic 56-isogeny field degree: | $2$ |
Cyclic 56-torsion field degree: | $48$ |
Full 56-torsion field degree: | $2304$ |
Jacobian
Conductor: | $2^{375}\cdot7^{163}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{21}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 224.2.b.a, 224.2.b.b, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g, 448.2.a.a, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.g, 448.2.a.h, 1568.2.b.f, 1568.2.b.g, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.h, 3136.2.a.j, 3136.2.a.s, 3136.2.a.u |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal 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 |
---|---|---|---|---|---|---|
$X_{\mathrm{sp}}^+(7)$ | $7$ | $48$ | $48$ | $0$ | $0$ | full Jacobian |
8.48.0.m.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.48.0.m.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
56.672.45.cr.1 | $56$ | $2$ | $2$ | $45$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
56.672.45.gq.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
56.672.45.gx.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.2688.185.ra.2 | $56$ | $2$ | $2$ | $185$ | $27$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rd.1 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rg.2 | $56$ | $2$ | $2$ | $185$ | $25$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rj.1 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rn.1 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rp.1 | $56$ | $2$ | $2$ | $185$ | $31$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rt.1 | $56$ | $2$ | $2$ | $185$ | $20$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.2688.185.rv.1 | $56$ | $2$ | $2$ | $185$ | $35$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
56.4032.277.ix.1 | $56$ | $3$ | $3$ | $277$ | $41$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |