Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
Genus: | $73 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $28^{12}\cdot56^{12}$ | Cusp orbits | $3^{2}\cdot6^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $9$ | ||||||
$\Q$-gonality: | $10 \le \gamma \le 32$ | ||||||
$\overline{\Q}$-gonality: | $10 \le \gamma \le 32$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2016.73.53 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&16\\42&13\end{bmatrix}$, $\begin{bmatrix}15&32\\26&53\end{bmatrix}$, $\begin{bmatrix}31&4\\6&25\end{bmatrix}$, $\begin{bmatrix}31&36\\10&9\end{bmatrix}$, $\begin{bmatrix}41&32\\24&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.1008.73.ee.2 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $192$ |
Full 56-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{300}\cdot7^{144}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{5}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
Newforms: | 64.2.a.a, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g, 1568.2.b.f, 1568.2.b.g, 3136.2.a.bd, 3136.2.a.bh, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bl, 3136.2.a.bo, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.bz |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,37,67,149$, 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 |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
8.96.1-8.m.2.2 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.96.1-8.m.2.2 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
56.1008.34-56.s.2.1 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
56.1008.34-56.s.2.31 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
56.1008.34-56.z.2.1 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
56.1008.34-56.z.2.28 | $56$ | $2$ | $2$ | $34$ | $1$ | $1\cdot2^{8}\cdot4\cdot6\cdot12$ |
56.1008.37-56.d.1.2 | $56$ | $2$ | $2$ | $37$ | $9$ | $6^{2}\cdot12^{2}$ |
56.1008.37-56.d.1.39 | $56$ | $2$ | $2$ | $37$ | $9$ | $6^{2}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.4032.145-56.gr.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.hc.2.5 | $56$ | $2$ | $2$ | $145$ | $33$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.jh.1.7 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.jw.2.5 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.ki.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.pd.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.pu.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.qe.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.qm.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.qs.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.re.2.1 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.rg.2.1 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.ru.2.1 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.rw.2.1 | $56$ | $2$ | $2$ | $145$ | $22$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.si.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.so.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{28}\cdot2^{8}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.sw.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.tg.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.tk.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.tr.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{12}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
56.4032.145-56.vt.1.5 | $56$ | $2$ | $2$ | $145$ | $27$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.we.1.5 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.xx.1.5 | $56$ | $2$ | $2$ | $145$ | $29$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |
56.4032.145-56.yi.1.5 | $56$ | $2$ | $2$ | $145$ | $18$ | $1^{22}\cdot2^{11}\cdot4^{4}\cdot6^{2}$ |