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Properties

Label 56.10752.401-56.kh.2.6
Level $56$
Index $10752$
Genus $401$
Analytic rank $59$
Cusps $96$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $10752$ $\PSL_2$-index:$5376$
Genus: $401 = 1 + \frac{ 5376 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 96 }{2}$
Cusps: $96$ (none of which are rational) Cusp widths $56^{96}$ Cusp orbits $2^{6}\cdot4^{3}\cdot6^{6}\cdot12^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $59$
$\Q$-gonality: $54 \le \gamma \le 112$
$\overline{\Q}$-gonality: $54 \le \gamma \le 112$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.10752.401.32

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}1&24\\48&41\end{bmatrix}$, $\begin{bmatrix}23&44\\14&33\end{bmatrix}$, $\begin{bmatrix}31&0\\0&3\end{bmatrix}$, $\begin{bmatrix}31&44\\46&25\end{bmatrix}$
$\GL_2(\Z/56\Z)$-subgroup: $C_6^2:D_4$
Contains $-I$: no $\quad$ (see 56.5376.401.kh.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $2$
Cyclic 56-torsion field degree: $48$
Full 56-torsion field degree: $288$

Jacobian

Conductor: $2^{1816}\cdot7^{687}$
Simple: no
Squarefree: no
Decomposition: $1^{69}\cdot2^{70}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$
Newforms: 14.2.a.a$^{6}$, 32.2.a.a$^{4}$, 56.2.a.a$^{4}$, 56.2.a.b$^{4}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 64.2.a.a$^{2}$, 64.2.b.a$^{2}$, 98.2.a.b$^{6}$, 112.2.a.a$^{3}$, 112.2.a.b$^{3}$, 112.2.a.c$^{3}$, 196.2.a.b$^{5}$, 196.2.a.c$^{5}$, 224.2.a.a$^{2}$, 224.2.a.b$^{2}$, 224.2.a.c$^{2}$, 224.2.a.d$^{2}$, 224.2.b.a$^{4}$, 224.2.b.b$^{4}$, 392.2.a.c$^{4}$, 392.2.a.f$^{4}$, 392.2.a.g$^{4}$, 392.2.b.e$^{2}$, 392.2.b.g$^{2}$, 448.2.a.a, 448.2.a.b, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.f, 448.2.a.g, 448.2.a.h, 448.2.a.i, 448.2.a.j, 448.2.b.a, 448.2.b.b, 448.2.b.c, 448.2.b.d, 784.2.a.a$^{3}$, 784.2.a.d$^{3}$, 784.2.a.h$^{3}$, 784.2.a.k$^{3}$, 784.2.a.l$^{3}$, 784.2.a.m$^{3}$, 1568.2.a.l$^{2}$, 1568.2.a.m$^{2}$, 1568.2.a.o$^{2}$, 1568.2.a.p$^{2}$, 1568.2.a.q$^{2}$, 1568.2.a.r$^{2}$, 1568.2.a.s$^{2}$, 1568.2.a.w$^{2}$, 1568.2.a.x$^{2}$, 1568.2.b.f$^{4}$, 1568.2.b.g$^{4}$, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bd, 3136.2.a.bh, 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.bv, 3136.2.a.bw, 3136.2.a.bz, 3136.2.a.h, 3136.2.a.j, 3136.2.a.s, 3136.2.a.u, 3136.2.b.e, 3136.2.b.f, 3136.2.b.j, 3136.2.b.l, 3136.2.b.n

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=31,79,223$, 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
$X_{\mathrm{sp}}^+(7)$ $7$ $384$ $192$ $0$ $0$ full Jacobian
8.384.5-8.b.3.4 $8$ $28$ $28$ $5$ $0$ $1^{66}\cdot2^{69}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.384.5-8.b.3.4 $8$ $28$ $28$ $5$ $0$ $1^{66}\cdot2^{69}\cdot4^{14}\cdot6^{6}\cdot12^{7}\cdot16$
56.5376.193-56.lc.2.8 $56$ $2$ $2$ $193$ $29$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.193-56.lc.2.21 $56$ $2$ $2$ $193$ $29$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.193-56.lj.1.3 $56$ $2$ $2$ $193$ $22$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.193-56.lj.1.21 $56$ $2$ $2$ $193$ $22$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.193-56.od.1.4 $56$ $2$ $2$ $193$ $22$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.193-56.od.1.14 $56$ $2$ $2$ $193$ $22$ $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$
56.5376.201-56.ba.1.6 $56$ $2$ $2$ $201$ $22$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$
56.5376.201-56.ba.1.24 $56$ $2$ $2$ $201$ $22$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$
56.5376.201-56.cg.2.3 $56$ $2$ $2$ $201$ $29$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$
56.5376.201-56.cg.2.32 $56$ $2$ $2$ $201$ $29$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$
56.5376.201-56.cn.2.7 $56$ $2$ $2$ $201$ $59$ $2^{10}\cdot4^{11}\cdot6^{6}\cdot12^{7}\cdot16$
56.5376.201-56.cn.2.14 $56$ $2$ $2$ $201$ $59$ $2^{10}\cdot4^{11}\cdot6^{6}\cdot12^{7}\cdot16$
56.5376.201-56.co.2.6 $56$ $2$ $2$ $201$ $22$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$
56.5376.201-56.co.2.31 $56$ $2$ $2$ $201$ $22$ $1^{32}\cdot2^{36}\cdot4^{6}\cdot6^{4}\cdot12^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.21504.801-56.fl.1.9 $56$ $2$ $2$ $801$ $129$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.gd.1.8 $56$ $2$ $2$ $801$ $153$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.kt.1.8 $56$ $2$ $2$ $801$ $127$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.ll.1.8 $56$ $2$ $2$ $801$ $149$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.ul.3.8 $56$ $2$ $2$ $801$ $142$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.ut.2.6 $56$ $2$ $2$ $801$ $135$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.wx.3.8 $56$ $2$ $2$ $801$ $138$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.21504.801-56.xf.2.6 $56$ $2$ $2$ $801$ $135$ $1^{134}\cdot2^{57}\cdot4^{24}\cdot6^{6}\cdot8\cdot12$
56.32256.1201-56.nd.3.8 $56$ $3$ $3$ $1201$ $203$ $1^{190}\cdot2^{123}\cdot4^{31}\cdot6^{18}\cdot8\cdot12^{9}\cdot16$