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Properties

Label 56.336.21.cw.1
Level $56$
Index $336$
Genus $21$
Analytic rank $5$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $784$
Index: $336$ $\PSL_2$-index:$336$
Genus: $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $2$ are rational) Cusp widths $7^{8}\cdot14^{4}\cdot56^{4}$ Cusp orbits $1^{2}\cdot2\cdot3^{2}\cdot6$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $5$
$\Q$-gonality: $6 \le \gamma \le 12$
$\overline{\Q}$-gonality: $6 \le \gamma \le 12$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 56F21
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.336.21.5

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}12&9\\25&0\end{bmatrix}$, $\begin{bmatrix}15&42\\42&15\end{bmatrix}$, $\begin{bmatrix}29&4\\28&13\end{bmatrix}$, $\begin{bmatrix}31&40\\10&29\end{bmatrix}$, $\begin{bmatrix}38&41\\19&32\end{bmatrix}$, $\begin{bmatrix}40&11\\15&16\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.672.21-56.cw.1.1, 56.672.21-56.cw.1.2, 56.672.21-56.cw.1.3, 56.672.21-56.cw.1.4, 56.672.21-56.cw.1.5, 56.672.21-56.cw.1.6, 56.672.21-56.cw.1.7, 56.672.21-56.cw.1.8, 56.672.21-56.cw.1.9, 56.672.21-56.cw.1.10, 56.672.21-56.cw.1.11, 56.672.21-56.cw.1.12, 56.672.21-56.cw.1.13, 56.672.21-56.cw.1.14, 56.672.21-56.cw.1.15, 56.672.21-56.cw.1.16, 56.672.21-56.cw.1.17, 56.672.21-56.cw.1.18, 56.672.21-56.cw.1.19, 56.672.21-56.cw.1.20, 56.672.21-56.cw.1.21, 56.672.21-56.cw.1.22, 56.672.21-56.cw.1.23, 56.672.21-56.cw.1.24, 56.672.21-56.cw.1.25, 56.672.21-56.cw.1.26, 56.672.21-56.cw.1.27, 56.672.21-56.cw.1.28, 56.672.21-56.cw.1.29, 56.672.21-56.cw.1.30, 56.672.21-56.cw.1.31, 56.672.21-56.cw.1.32
Cyclic 56-isogeny field degree: $4$
Cyclic 56-torsion field degree: $96$
Full 56-torsion field degree: $9216$

Jacobian

Conductor: $2^{60}\cdot7^{37}$
Simple: no
Squarefree: no
Decomposition: $1^{9}\cdot2^{6}$
Newforms: 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal 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
$X_{\mathrm{sp}}^+(7)$ $7$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.o.1 $8$ $28$ $28$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.o.1 $8$ $28$ $28$ $0$ $0$ full Jacobian
28.168.9.c.1 $28$ $2$ $2$ $9$ $0$ $1^{6}\cdot2^{3}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.672.41.ns.1 $56$ $2$ $2$ $41$ $10$ $1^{18}\cdot2$
56.672.41.nu.1 $56$ $2$ $2$ $41$ $12$ $1^{18}\cdot2$
56.672.41.oa.1 $56$ $2$ $2$ $41$ $14$ $1^{18}\cdot2$
56.672.41.oc.1 $56$ $2$ $2$ $41$ $15$ $1^{18}\cdot2$
56.672.41.oy.1 $56$ $2$ $2$ $41$ $10$ $1^{18}\cdot2$
56.672.41.pa.1 $56$ $2$ $2$ $41$ $14$ $1^{18}\cdot2$
56.672.41.pg.1 $56$ $2$ $2$ $41$ $12$ $1^{18}\cdot2$
56.672.41.pi.1 $56$ $2$ $2$ $41$ $13$ $1^{18}\cdot2$
56.672.45.cj.1 $56$ $2$ $2$ $45$ $7$ $1^{12}\cdot2^{6}$
56.672.45.ck.1 $56$ $2$ $2$ $45$ $7$ $1^{12}\cdot2^{6}$
56.672.45.ey.1 $56$ $2$ $2$ $45$ $15$ $1^{12}\cdot2^{6}$
56.672.45.fa.1 $56$ $2$ $2$ $45$ $25$ $1^{12}\cdot2^{6}$
56.672.45.fs.1 $56$ $2$ $2$ $45$ $11$ $1^{12}\cdot2^{6}$
56.672.45.fu.1 $56$ $2$ $2$ $45$ $17$ $1^{12}\cdot2^{6}$
56.672.45.ge.1 $56$ $2$ $2$ $45$ $13$ $1^{12}\cdot2^{6}$
56.672.45.gg.1 $56$ $2$ $2$ $45$ $17$ $1^{12}\cdot2^{6}$
56.672.45.hs.1 $56$ $2$ $2$ $45$ $13$ $1^{20}\cdot2^{2}$
56.672.45.hu.1 $56$ $2$ $2$ $45$ $15$ $1^{20}\cdot2^{2}$
56.672.45.ia.1 $56$ $2$ $2$ $45$ $19$ $1^{20}\cdot2^{2}$
56.672.45.ic.1 $56$ $2$ $2$ $45$ $9$ $1^{20}\cdot2^{2}$
56.672.45.iy.1 $56$ $2$ $2$ $45$ $7$ $1^{20}\cdot2^{2}$
56.672.45.ja.1 $56$ $2$ $2$ $45$ $13$ $1^{20}\cdot2^{2}$
56.672.45.jg.1 $56$ $2$ $2$ $45$ $25$ $1^{20}\cdot2^{2}$
56.672.45.ji.1 $56$ $2$ $2$ $45$ $11$ $1^{20}\cdot2^{2}$
56.1008.61.ig.1 $56$ $3$ $3$ $61$ $16$ $1^{26}\cdot2^{7}$