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Properties

Label 56.10752.385-56.qx.2.8
Level $56$
Index $10752$
Genus $385$
Analytic rank $66$
Cusps $128$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $10752$ $\PSL_2$-index:$5376$
Genus: $385 = 1 + \frac{ 5376 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 128 }{2}$
Cusps: $128$ (none of which are rational) Cusp widths $28^{64}\cdot56^{64}$ Cusp orbits $4^{6}\cdot6^{12}\cdot8\cdot12^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $66$
$\Q$-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: 56.10752.385.25864

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}15&18\\30&55\end{bmatrix}$, $\begin{bmatrix}19&6\\52&1\end{bmatrix}$, $\begin{bmatrix}27&14\\52&1\end{bmatrix}$, $\begin{bmatrix}41&48\\10&23\end{bmatrix}$
$\GL_2(\Z/56\Z)$-subgroup: $C_6^2:C_2^3$
Contains $-I$: no $\quad$ (see 56.5376.385.qx.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $4$
Cyclic 56-torsion field degree: $24$
Full 56-torsion field degree: $288$

Jacobian

Conductor: $2^{1601}\cdot7^{670}$
Simple: no
Squarefree: no
Decomposition: $1^{107}\cdot2^{57}\cdot4^{17}\cdot6^{8}\cdot12^{4}$
Newforms: 14.2.a.a$^{6}$, 32.2.a.a$^{3}$, 49.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{6}$, 56.2.b.b$^{6}$, 98.2.a.a$^{3}$, 98.2.a.b$^{3}$, 196.2.a.b$^{4}$, 196.2.a.c$^{2}$, 224.2.a.a$^{2}$, 224.2.a.b$^{2}$, 224.2.a.c$^{2}$, 224.2.a.d$^{2}$, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 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$^{3}$, 392.2.b.b$^{3}$, 392.2.b.c$^{3}$, 392.2.b.d$^{3}$, 392.2.b.e$^{6}$, 392.2.b.g$^{3}$, 448.2.a.a$^{4}$, 448.2.a.c$^{4}$, 448.2.a.d$^{4}$, 448.2.a.e$^{4}$, 448.2.a.g$^{4}$, 448.2.a.h$^{4}$, 1568.2.a.a, 1568.2.a.b, 1568.2.a.c, 1568.2.a.d, 1568.2.a.e, 1568.2.a.f, 1568.2.a.g, 1568.2.a.h, 1568.2.a.i, 1568.2.a.k, 1568.2.a.l$^{2}$, 1568.2.a.m$^{2}$, 1568.2.a.o, 1568.2.a.p, 1568.2.a.q, 1568.2.a.r, 1568.2.a.s$^{2}$, 1568.2.a.v, 1568.2.a.w$^{2}$, 1568.2.a.x, 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, 3136.2.a.b$^{4}$, 3136.2.a.bc$^{4}$, 3136.2.a.bk$^{2}$, 3136.2.a.bm$^{2}$, 3136.2.a.bn$^{2}$, 3136.2.a.bp$^{2}$, 3136.2.a.bq$^{2}$, 3136.2.a.br$^{2}$, 3136.2.a.bs$^{2}$, 3136.2.a.bt$^{2}$, 3136.2.a.c$^{2}$, 3136.2.a.e$^{2}$, 3136.2.a.h$^{4}$, 3136.2.a.j$^{4}$, 3136.2.a.n$^{2}$, 3136.2.a.o$^{2}$, 3136.2.a.p$^{2}$, 3136.2.a.q$^{2}$, 3136.2.a.s$^{4}$, 3136.2.a.u$^{4}$, 3136.2.a.w$^{2}$, 3136.2.a.z$^{2}$

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,13,\ldots,2273$, 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
7.56.1.b.1 $7$ $192$ $96$ $1$ $0$ $1^{106}\cdot2^{57}\cdot4^{17}\cdot6^{8}\cdot12^{4}$
8.192.1-8.h.1.4 $8$ $56$ $56$ $1$ $0$ $1^{106}\cdot2^{57}\cdot4^{17}\cdot6^{8}\cdot12^{4}$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.5376.185-56.bv.2.1 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.bv.2.15 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.bx.1.1 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.bx.1.16 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.pe.2.9 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.pe.2.21 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.pg.1.5 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.185-56.pg.1.23 $56$ $2$ $2$ $185$ $25$ $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.iv.2.6 $56$ $2$ $2$ $193$ $22$ $1^{64}\cdot2^{24}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.iv.2.24 $56$ $2$ $2$ $193$ $22$ $1^{64}\cdot2^{24}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.iz.1.8 $56$ $2$ $2$ $193$ $66$ $2^{16}\cdot4^{16}\cdot6^{8}\cdot12^{4}$
56.5376.193-56.iz.1.17 $56$ $2$ $2$ $193$ $66$ $2^{16}\cdot4^{16}\cdot6^{8}\cdot12^{4}$
56.5376.193-56.jb.2.9 $56$ $2$ $2$ $193$ $22$ $1^{64}\cdot2^{24}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.jb.2.16 $56$ $2$ $2$ $193$ $22$ $1^{64}\cdot2^{24}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.nf.1.8 $56$ $2$ $2$ $193$ $32$ $1^{54}\cdot2^{29}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.nf.1.10 $56$ $2$ $2$ $193$ $32$ $1^{54}\cdot2^{29}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.nj.1.4 $56$ $2$ $2$ $193$ $32$ $1^{70}\cdot2^{17}\cdot4^{4}\cdot6^{4}\cdot12^{4}$
56.5376.193-56.nj.1.15 $56$ $2$ $2$ $193$ $32$ $1^{70}\cdot2^{17}\cdot4^{4}\cdot6^{4}\cdot12^{4}$
56.5376.193-56.qh.1.6 $56$ $2$ $2$ $193$ $32$ $1^{54}\cdot2^{29}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.qh.1.12 $56$ $2$ $2$ $193$ $32$ $1^{54}\cdot2^{29}\cdot4^{8}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.ql.2.6 $56$ $2$ $2$ $193$ $32$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.193-56.ql.2.12 $56$ $2$ $2$ $193$ $32$ $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$
56.5376.193-56.sp.1.4 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.sp.1.13 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.sx.1.5 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.sx.1.12 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.btm.1.4 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.btm.1.13 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.btu.1.3 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$
56.5376.193-56.btu.1.14 $56$ $2$ $2$ $193$ $33$ $1^{54}\cdot2^{27}\cdot4^{9}\cdot6^{4}\cdot12^{2}$

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.ft.2.8 $56$ $2$ $2$ $769$ $70$ $2^{28}\cdot4^{28}\cdot8^{17}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.fv.2.8 $56$ $2$ $2$ $769$ $74$ $2^{28}\cdot4^{28}\cdot8^{17}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.ob.2.8 $56$ $2$ $2$ $769$ $70$ $2^{28}\cdot4^{28}\cdot8^{17}\cdot12^{4}\cdot16^{2}$
56.21504.769-56.od.2.8 $56$ $2$ $2$ $769$ $74$ $2^{28}\cdot4^{28}\cdot8^{17}\cdot12^{4}\cdot16^{2}$
56.32256.1153-56.in.2.10 $56$ $3$ $3$ $1153$ $216$ $1^{204}\cdot2^{110}\cdot4^{26}\cdot6^{24}\cdot12^{8}$