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Properties

Label 56.4032.289.bmw.2
Level $56$
Index $4032$
Genus $289$
Analytic rank $54$
Cusps $96$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $4032$ $\PSL_2$-index:$4032$
Genus: $289 = 1 + \frac{ 4032 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 96 }{2}$
Cusps: $96$ (none of which are rational) Cusp widths $28^{48}\cdot56^{48}$ Cusp orbits $12^{6}\cdot24$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $54$
$\Q$-gonality: $40 \le \gamma \le 84$
$\overline{\Q}$-gonality: $40 \le \gamma \le 84$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.4032.289.4515

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}7&16\\54&49\end{bmatrix}$, $\begin{bmatrix}25&0\\42&11\end{bmatrix}$, $\begin{bmatrix}39&34\\6&15\end{bmatrix}$, $\begin{bmatrix}55&8\\8&23\end{bmatrix}$
$\GL_2(\Z/56\Z)$-subgroup: Group 768.26500
Contains $-I$: yes
Quadratic refinements: 56.8064.289-56.bmw.2.1, 56.8064.289-56.bmw.2.2, 56.8064.289-56.bmw.2.3, 56.8064.289-56.bmw.2.4, 56.8064.289-56.bmw.2.5, 56.8064.289-56.bmw.2.6, 56.8064.289-56.bmw.2.7, 56.8064.289-56.bmw.2.8
Cyclic 56-isogeny field degree: $8$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $768$

Jacobian

Conductor: $2^{1256}\cdot7^{555}$
Simple: no
Squarefree: no
Decomposition: $1^{67}\cdot2^{45}\cdot4^{9}\cdot6^{8}\cdot12^{4}$
Newforms: 14.2.a.a, 49.2.a.a, 56.2.a.a, 56.2.a.b, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.a, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 392.2.a.a, 392.2.a.c$^{2}$, 392.2.a.e, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 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.f$^{2}$, 392.2.b.g$^{3}$, 448.2.a.a$^{3}$, 448.2.a.e, 448.2.a.h, 784.2.a.a, 784.2.a.c, 784.2.a.d, 784.2.a.f, 784.2.a.g, 784.2.a.h, 784.2.a.j, 784.2.a.k, 784.2.a.l, 784.2.a.m, 784.2.a.n, 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.j, 1568.2.a.k, 1568.2.a.n, 1568.2.a.t, 1568.2.a.u, 1568.2.a.v, 1568.2.b.a, 1568.2.b.b$^{3}$, 1568.2.b.c$^{3}$, 1568.2.b.d, 1568.2.b.e$^{2}$, 1568.2.b.f$^{2}$, 1568.2.b.g, 3136.2.a.a, 3136.2.a.b, 3136.2.a.bb, 3136.2.a.bc$^{2}$, 3136.2.a.be, 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.bq$^{2}$, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bt, 3136.2.a.bu, 3136.2.a.bx, 3136.2.a.bz, 3136.2.a.c, 3136.2.a.e, 3136.2.a.h, 3136.2.a.i, 3136.2.a.j$^{2}$, 3136.2.a.k, 3136.2.a.m, 3136.2.a.n, 3136.2.a.o$^{5}$, 3136.2.a.p, 3136.2.a.q, 3136.2.a.s$^{3}$, 3136.2.a.t, 3136.2.a.u, 3136.2.a.v, 3136.2.a.w, 3136.2.a.z

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3,11,\ldots,743$, 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
56.2016.139.fm.1 $56$ $2$ $2$ $139$ $22$ $1^{30}\cdot2^{26}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.139.fn.1 $56$ $2$ $2$ $139$ $22$ $1^{30}\cdot2^{26}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.139.nh.1 $56$ $2$ $2$ $139$ $22$ $1^{30}\cdot2^{26}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.139.nk.1 $56$ $2$ $2$ $139$ $22$ $1^{30}\cdot2^{26}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.145.bgw.2 $56$ $2$ $2$ $145$ $26$ $1^{40}\cdot2^{20}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.2016.145.bgx.2 $56$ $2$ $2$ $145$ $26$ $1^{40}\cdot2^{20}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.2016.145.bic.2 $56$ $2$ $2$ $145$ $20$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.145.bie.1 $56$ $2$ $2$ $145$ $20$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.145.bis.1 $56$ $2$ $2$ $145$ $24$ $1^{34}\cdot2^{23}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.2016.145.biu.2 $56$ $2$ $2$ $145$ $24$ $1^{34}\cdot2^{23}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.2016.145.bkm.1 $56$ $2$ $2$ $145$ $24$ $1^{50}\cdot2^{11}\cdot6^{4}\cdot12^{4}$
56.2016.145.bks.2 $56$ $2$ $2$ $145$ $24$ $1^{50}\cdot2^{19}\cdot4^{8}\cdot6^{4}$
56.2016.145.bmk.1 $56$ $2$ $2$ $145$ $54$ $2^{8}\cdot4^{8}\cdot6^{8}\cdot12^{4}$
56.2016.145.bsy.2 $56$ $2$ $2$ $145$ $28$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.2016.145.bte.1 $56$ $2$ $2$ $145$ $28$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$