Properties

Label 56.504.34.z.1
Level $56$
Index $504$
Genus $34$
Analytic rank $1$
Cusps $18$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $392$
Index: $504$ $\PSL_2$-index:$504$
Genus: $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$
Cusps: $18$ (none of which are rational) Cusp widths $14^{6}\cdot28^{9}\cdot56^{3}$ Cusp orbits $3^{4}\cdot6$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $5 \le \gamma \le 16$
$\overline{\Q}$-gonality: $5 \le \gamma \le 16$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.504.34.1

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}3&0\\14&45\end{bmatrix}$, $\begin{bmatrix}11&32\\26&45\end{bmatrix}$, $\begin{bmatrix}15&48\\2&13\end{bmatrix}$, $\begin{bmatrix}15&48\\44&55\end{bmatrix}$, $\begin{bmatrix}17&8\\24&9\end{bmatrix}$, $\begin{bmatrix}29&20\\32&9\end{bmatrix}$, $\begin{bmatrix}31&28\\46&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.1008.34-56.z.1.1, 56.1008.34-56.z.1.2, 56.1008.34-56.z.1.3, 56.1008.34-56.z.1.4, 56.1008.34-56.z.1.5, 56.1008.34-56.z.1.6, 56.1008.34-56.z.1.7, 56.1008.34-56.z.1.8, 56.1008.34-56.z.1.9, 56.1008.34-56.z.1.10, 56.1008.34-56.z.1.11, 56.1008.34-56.z.1.12, 56.1008.34-56.z.1.13, 56.1008.34-56.z.1.14, 56.1008.34-56.z.1.15, 56.1008.34-56.z.1.16, 56.1008.34-56.z.1.17, 56.1008.34-56.z.1.18, 56.1008.34-56.z.1.19, 56.1008.34-56.z.1.20, 56.1008.34-56.z.1.21, 56.1008.34-56.z.1.22, 56.1008.34-56.z.1.23, 56.1008.34-56.z.1.24, 56.1008.34-56.z.1.25, 56.1008.34-56.z.1.26, 56.1008.34-56.z.1.27, 56.1008.34-56.z.1.28, 56.1008.34-56.z.1.29, 56.1008.34-56.z.1.30, 56.1008.34-56.z.1.31, 56.1008.34-56.z.1.32, 56.1008.34-56.z.1.33, 56.1008.34-56.z.1.34, 56.1008.34-56.z.1.35, 56.1008.34-56.z.1.36, 56.1008.34-56.z.1.37, 56.1008.34-56.z.1.38, 56.1008.34-56.z.1.39, 56.1008.34-56.z.1.40, 56.1008.34-56.z.1.41, 56.1008.34-56.z.1.42, 56.1008.34-56.z.1.43, 56.1008.34-56.z.1.44, 56.1008.34-56.z.1.45, 56.1008.34-56.z.1.46, 56.1008.34-56.z.1.47, 56.1008.34-56.z.1.48, 56.1008.34-56.z.1.49, 56.1008.34-56.z.1.50, 56.1008.34-56.z.1.51, 56.1008.34-56.z.1.52, 56.1008.34-56.z.1.53, 56.1008.34-56.z.1.54, 56.1008.34-56.z.1.55, 56.1008.34-56.z.1.56, 56.1008.34-56.z.1.57, 56.1008.34-56.z.1.58, 56.1008.34-56.z.1.59, 56.1008.34-56.z.1.60, 56.1008.34-56.z.1.61, 56.1008.34-56.z.1.62, 56.1008.34-56.z.1.63, 56.1008.34-56.z.1.64
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $6144$

Jacobian

Conductor: $2^{84}\cdot7^{68}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{6}\cdot6\cdot12$
Newforms: 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g

Rational points

This modular curve has no $\Q_p$ points for $p=3,11,67$, 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
$X_{\mathrm{ns}}^+(7)$ $7$ $24$ $24$ $0$ $0$ full Jacobian
8.24.0.e.2 $8$ $21$ $21$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.e.2 $8$ $21$ $21$ $0$ $0$ full Jacobian
28.252.16.b.1 $28$ $2$ $2$ $16$ $1$ $6\cdot12$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.1008.67.en.1 $56$ $2$ $2$ $67$ $5$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.er.1 $56$ $2$ $2$ $67$ $16$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.ev.2 $56$ $2$ $2$ $67$ $10$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.ez.2 $56$ $2$ $2$ $67$ $1$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.ff.2 $56$ $2$ $2$ $67$ $2$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.fj.1 $56$ $2$ $2$ $67$ $8$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.fn.2 $56$ $2$ $2$ $67$ $8$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.67.fr.2 $56$ $2$ $2$ $67$ $6$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.1008.70.h.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.l.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.t.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.x.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.bd.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.bf.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.bl.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.bn.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.bt.1 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.bv.1 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cb.1 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cd.1 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cj.1 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cl.1 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cr.1 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.ct.1 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.cz.2 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.db.2 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dh.2 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dj.2 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dp.2 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dr.2 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dx.2 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.dz.2 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.1008.70.eh.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.ej.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.ep.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.er.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.ez.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.fd.1 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.fl.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.70.fp.1 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.73.df.1 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.dn.1 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.ee.1 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.ei.1 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.gv.1 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.gx.1 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.hd.1 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.hf.1 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.1008.73.jt.1 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.jv.1 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kb.1 $56$ $2$ $2$ $73$ $8$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kd.1 $56$ $2$ $2$ $73$ $8$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kj.1 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kl.1 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kr.1 $56$ $2$ $2$ $73$ $10$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.1008.73.kt.1 $56$ $2$ $2$ $73$ $10$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$