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Properties

Label 56.2688.193.nx.2
Level $56$
Index $2688$
Genus $193$
Analytic rank $31$
Cusps $64$
$\Q$-cusps $2$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $2688$ $\PSL_2$-index:$2688$
Genus: $193 = 1 + \frac{ 2688 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 64 }{2}$
Cusps: $64$ (of which $2$ are rational) Cusp widths $28^{32}\cdot56^{32}$ Cusp orbits $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}\cdot8\cdot24$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $31$
$\Q$-gonality: $27 \le \gamma \le 56$
$\overline{\Q}$-gonality: $27 \le \gamma \le 56$
Rational cusps: $2$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.2688.193.457

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}23&12\\10&21\end{bmatrix}$, $\begin{bmatrix}25&44\\0&45\end{bmatrix}$, $\begin{bmatrix}33&4\\32&9\end{bmatrix}$, $\begin{bmatrix}39&36\\16&47\end{bmatrix}$, $\begin{bmatrix}47&20\\0&51\end{bmatrix}$, $\begin{bmatrix}47&28\\14&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.5376.193-56.nx.2.1, 56.5376.193-56.nx.2.2, 56.5376.193-56.nx.2.3, 56.5376.193-56.nx.2.4, 56.5376.193-56.nx.2.5, 56.5376.193-56.nx.2.6, 56.5376.193-56.nx.2.7, 56.5376.193-56.nx.2.8, 56.5376.193-56.nx.2.9, 56.5376.193-56.nx.2.10, 56.5376.193-56.nx.2.11, 56.5376.193-56.nx.2.12, 56.5376.193-56.nx.2.13, 56.5376.193-56.nx.2.14, 56.5376.193-56.nx.2.15, 56.5376.193-56.nx.2.16, 56.5376.193-56.nx.2.17, 56.5376.193-56.nx.2.18, 56.5376.193-56.nx.2.19, 56.5376.193-56.nx.2.20, 56.5376.193-56.nx.2.21, 56.5376.193-56.nx.2.22, 56.5376.193-56.nx.2.23, 56.5376.193-56.nx.2.24, 56.5376.193-56.nx.2.25, 56.5376.193-56.nx.2.26, 56.5376.193-56.nx.2.27, 56.5376.193-56.nx.2.28, 56.5376.193-56.nx.2.29, 56.5376.193-56.nx.2.30, 56.5376.193-56.nx.2.31, 56.5376.193-56.nx.2.32
Cyclic 56-isogeny field degree: $4$
Cyclic 56-torsion field degree: $96$
Full 56-torsion field degree: $1152$

Jacobian

Conductor: $2^{810}\cdot7^{363}$
Simple: no
Squarefree: no
Decomposition: $1^{37}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
Newforms: 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a, 56.2.b.b, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a, 224.2.b.b, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.b, 392.2.b.c, 392.2.b.e, 392.2.b.f, 392.2.b.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.a, 1568.2.b.d, 1568.2.b.e, 1568.2.b.f, 1568.2.b.g$^{2}$, 3136.2.a.a, 3136.2.a.bb, 3136.2.a.be, 3136.2.a.bf, 3136.2.a.bg, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bk, 3136.2.a.bl, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bo, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bu, 3136.2.a.bx, 3136.2.a.by, 3136.2.a.bz, 3136.2.a.c, 3136.2.a.e, 3136.2.a.f, 3136.2.a.i, 3136.2.a.k, 3136.2.a.m$^{2}$, 3136.2.a.p, 3136.2.a.q, 3136.2.a.t, 3136.2.a.v, 3136.2.a.w, 3136.2.a.y, 3136.2.a.z

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.96.1.n.2 $56$ $28$ $28$ $1$ $1$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
56.1344.93.g.1 $56$ $2$ $2$ $93$ $7$ $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.1344.93.h.2 $56$ $2$ $2$ $93$ $7$ $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
56.1344.97.ct.1 $56$ $2$ $2$ $97$ $31$ $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.5376.385.il.1 $56$ $2$ $2$ $385$ $67$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.jz.2 $56$ $2$ $2$ $385$ $75$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.pn.2 $56$ $2$ $2$ $385$ $65$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.rb.1 $56$ $2$ $2$ $385$ $71$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.zj.1 $56$ $2$ $2$ $385$ $64$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.bad.2 $56$ $2$ $2$ $385$ $73$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.bdn.2 $56$ $2$ $2$ $385$ $60$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.385.beh.1 $56$ $2$ $2$ $385$ $73$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.401.kl.1 $56$ $2$ $2$ $401$ $77$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.kr.2 $56$ $2$ $2$ $401$ $65$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.lh.2 $56$ $2$ $2$ $401$ $77$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.lo.1 $56$ $2$ $2$ $401$ $65$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.nl.2 $56$ $2$ $2$ $401$ $65$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.nm.1 $56$ $2$ $2$ $401$ $81$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.ou.1 $56$ $2$ $2$ $401$ $65$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.ov.2 $56$ $2$ $2$ $401$ $81$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.qa.2 $56$ $2$ $2$ $401$ $65$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.qb.1 $56$ $2$ $2$ $401$ $81$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.rd.1 $56$ $2$ $2$ $401$ $65$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.re.2 $56$ $2$ $2$ $401$ $81$ $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$
56.5376.401.rl.1 $56$ $2$ $2$ $401$ $77$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.rs.3 $56$ $2$ $2$ $401$ $65$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.tb.2 $56$ $2$ $2$ $401$ $77$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.401.th.1 $56$ $2$ $2$ $401$ $65$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.8064.577.rb.2 $56$ $3$ $3$ $577$ $101$ $1^{102}\cdot2^{55}\cdot4^{13}\cdot6^{12}\cdot12^{4}$