Properties

Label 56.1008.34-56.y.2.40
Level $56$
Index $1008$
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: $1008$ $\PSL_2$-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 $3^{2}\cdot6^{2}$
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: 56.1008.34.2003

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}11&44\\52&45\end{bmatrix}$, $\begin{bmatrix}15&16\\10&39\end{bmatrix}$, $\begin{bmatrix}15&24\\18&13\end{bmatrix}$, $\begin{bmatrix}27&36\\4&15\end{bmatrix}$, $\begin{bmatrix}31&40\\32&49\end{bmatrix}$, $\begin{bmatrix}31&52\\8&39\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.504.34.y.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $3072$

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.f, 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

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.48.0-56.i.1.6 $56$ $21$ $21$ $0$ $0$ full Jacobian
56.504.16-28.b.1.10 $56$ $2$ $2$ $16$ $1$ $6\cdot12$
56.504.16-28.b.1.40 $56$ $2$ $2$ $16$ $1$ $6\cdot12$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.2016.67-56.em.2.24 $56$ $2$ $2$ $67$ $5$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.eq.2.20 $56$ $2$ $2$ $67$ $16$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.eu.2.24 $56$ $2$ $2$ $67$ $10$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.ey.2.24 $56$ $2$ $2$ $67$ $1$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fe.2.34 $56$ $2$ $2$ $67$ $2$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fi.2.20 $56$ $2$ $2$ $67$ $8$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fm.2.24 $56$ $2$ $2$ $67$ $8$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.67-56.fq.2.24 $56$ $2$ $2$ $67$ $6$ $1^{13}\cdot2^{3}\cdot4^{2}\cdot6$
56.2016.70-56.g.1.32 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.k.1.36 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.s.2.18 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.w.1.31 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ba.2.20 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bf.1.32 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bi.1.31 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bn.2.19 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.bs.2.11 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.bu.2.16 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.ca.2.15 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cc.2.14 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.ci.2.16 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.ck.2.10 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cq.2.15 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cs.2.15 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.cy.1.22 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.da.1.22 $56$ $2$ $2$ $70$ $9$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dg.1.22 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.di.1.22 $56$ $2$ $2$ $70$ $6$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.do.1.23 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dq.1.23 $56$ $2$ $2$ $70$ $10$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dw.1.23 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.dy.1.23 $56$ $2$ $2$ $70$ $7$ $1^{14}\cdot2^{4}\cdot4^{2}\cdot6$
56.2016.70-56.ee.2.7 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ej.2.7 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.em.2.2 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.er.1.8 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.ey.2.6 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fc.1.7 $56$ $2$ $2$ $70$ $10$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fk.1.8 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.70-56.fo.2.3 $56$ $2$ $2$ $70$ $6$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.2016.73-56.de.1.32 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.dm.2.22 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.ed.2.32 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.eh.1.32 $56$ $2$ $2$ $73$ $9$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.gs.2.32 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.gx.1.32 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.ha.1.32 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.hf.2.32 $56$ $2$ $2$ $73$ $10$ $1\cdot2^{8}\cdot4\cdot6\cdot12$
56.2016.73-56.js.2.22 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.ju.2.23 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.ka.2.24 $56$ $2$ $2$ $73$ $8$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kc.2.19 $56$ $2$ $2$ $73$ $8$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.ki.2.23 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kk.2.23 $56$ $2$ $2$ $73$ $12$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.kq.2.18 $56$ $2$ $2$ $73$ $10$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$
56.2016.73-56.ks.2.24 $56$ $2$ $2$ $73$ $10$ $1^{9}\cdot2^{8}\cdot4^{2}\cdot6$