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Properties

Label 56.10752.369-56.iv.2.16
Level $56$
Index $10752$
Genus $369$
Analytic rank $27$
Cusps $160$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $10752$ $\PSL_2$-index:$5376$
Genus: $369 = 1 + \frac{ 5376 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 160 }{2}$
Cusps: $160$ (none of which are rational) Cusp widths $14^{64}\cdot28^{32}\cdot56^{64}$ Cusp orbits $4^{2}\cdot6^{2}\cdot8^{4}\cdot12^{9}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $27$
$\Q$-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: 56.10752.369.8596

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}3&28\\8&25\end{bmatrix}$, $\begin{bmatrix}11&52\\24&33\end{bmatrix}$, $\begin{bmatrix}29&14\\40&55\end{bmatrix}$, $\begin{bmatrix}51&0\\28&23\end{bmatrix}$
$\GL_2(\Z/56\Z)$-subgroup: $C_6^2:C_2^3$
Contains $-I$: no $\quad$ (see 56.5376.369.iv.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $4$
Cyclic 56-torsion field degree: $48$
Full 56-torsion field degree: $288$

Jacobian

Conductor: $2^{1617}\cdot7^{648}$
Simple: no
Squarefree: no
Decomposition: $1^{59}\cdot2^{39}\cdot4^{24}\cdot6^{4}\cdot8^{8}\cdot12^{4}$
Newforms: 14.2.a.a$^{6}$, 28.2.d.a$^{2}$, 49.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 98.2.a.a$^{3}$, 98.2.a.b$^{3}$, 112.2.f.a$^{2}$, 112.2.f.b$^{2}$, 196.2.a.b$^{4}$, 196.2.a.c$^{2}$, 196.2.d.a, 196.2.d.b, 196.2.d.c, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 224.2.e.a$^{4}$, 224.2.e.b$^{4}$, 392.2.a.b, 392.2.a.c$^{2}$, 392.2.a.d, 392.2.a.f$^{2}$, 392.2.a.g, 392.2.a.h, 392.2.b.a, 392.2.b.b, 392.2.b.c, 392.2.b.d, 392.2.b.e$^{2}$, 392.2.b.g, 448.2.a.a$^{2}$, 448.2.a.c$^{2}$, 448.2.a.d$^{2}$, 448.2.a.e$^{2}$, 448.2.a.g$^{2}$, 448.2.a.h$^{2}$, 448.2.f.b$^{4}$, 784.2.f.a, 784.2.f.b, 784.2.f.c, 784.2.f.d, 784.2.f.e, 1568.2.b.a, 1568.2.b.b, 1568.2.b.c, 1568.2.b.d, 1568.2.b.f$^{2}$, 1568.2.b.g, 1568.2.e.a$^{2}$, 1568.2.e.b$^{2}$, 1568.2.e.c$^{2}$, 1568.2.e.d$^{2}$, 1568.2.e.e$^{2}$, 3136.2.a.b$^{2}$, 3136.2.a.bc$^{2}$, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.bq, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bt, 3136.2.a.c, 3136.2.a.e, 3136.2.a.h$^{2}$, 3136.2.a.j$^{2}$, 3136.2.a.n, 3136.2.a.o, 3136.2.a.p, 3136.2.a.q, 3136.2.a.s$^{2}$, 3136.2.a.u$^{2}$, 3136.2.a.w, 3136.2.a.z, 3136.2.f.c$^{2}$, 3136.2.f.e$^{2}$, 3136.2.f.i$^{2}$

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,11,\ldots,443$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.5376.177-56.da.1.23 $56$ $2$ $2$ $177$ $3$ $1^{32}\cdot2^{22}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.177-56.da.1.28 $56$ $2$ $2$ $177$ $3$ $1^{32}\cdot2^{22}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.177-56.dd.2.24 $56$ $2$ $2$ $177$ $3$ $1^{32}\cdot2^{22}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.177-56.dd.2.29 $56$ $2$ $2$ $177$ $3$ $1^{32}\cdot2^{22}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.177-56.eg.2.11 $56$ $2$ $2$ $177$ $27$ $2^{12}\cdot4^{16}\cdot6^{4}\cdot8^{4}\cdot12^{4}$
56.5376.177-56.eg.2.20 $56$ $2$ $2$ $177$ $27$ $2^{12}\cdot4^{16}\cdot6^{4}\cdot8^{4}\cdot12^{4}$
56.5376.185-56.ii.1.16 $56$ $2$ $2$ $185$ $5$ $1^{32}\cdot2^{18}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.185-56.ii.1.27 $56$ $2$ $2$ $185$ $5$ $1^{32}\cdot2^{18}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.185-56.il.2.23 $56$ $2$ $2$ $185$ $5$ $1^{32}\cdot2^{18}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.185-56.il.2.28 $56$ $2$ $2$ $185$ $5$ $1^{32}\cdot2^{18}\cdot4^{12}\cdot6^{2}\cdot8^{4}\cdot12^{2}$
56.5376.185-56.pe.2.1 $56$ $2$ $2$ $185$ $25$ $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$
56.5376.185-56.pe.2.21 $56$ $2$ $2$ $185$ $25$ $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$
56.5376.185-56.py.1.14 $56$ $2$ $2$ $185$ $25$ $2^{20}\cdot4^{16}\cdot6^{4}\cdot8^{4}\cdot12^{2}$
56.5376.185-56.py.1.24 $56$ $2$ $2$ $185$ $25$ $2^{20}\cdot4^{16}\cdot6^{4}\cdot8^{4}\cdot12^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.21504.769-56.me.2.8 $56$ $2$ $2$ $769$ $69$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.ml.2.7 $56$ $2$ $2$ $769$ $70$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.nu.2.1 $56$ $2$ $2$ $769$ $69$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.ob.2.8 $56$ $2$ $2$ $769$ $70$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.tp.2.4 $56$ $2$ $2$ $769$ $71$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.tq.2.3 $56$ $2$ $2$ $769$ $74$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.tt.2.3 $56$ $2$ $2$ $769$ $71$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.tu.2.4 $56$ $2$ $2$ $769$ $74$ $1^{48}\cdot2^{46}\cdot4^{21}\cdot6^{4}\cdot8^{9}\cdot12^{4}\cdot16^{2}$
56.32256.1105-56.ip.2.16 $56$ $3$ $3$ $1105$ $93$ $1^{116}\cdot2^{62}\cdot4^{44}\cdot6^{12}\cdot8^{16}\cdot12^{10}$