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Properties

Label 56.768.49.er.2
Level $56$
Index $768$
Genus $49$
Analytic rank $8$
Cusps $32$
$\Q$-cusps $4$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $768$ $\PSL_2$-index:$768$
Genus: $49 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$
Cusps: $32$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{8}\cdot28^{8}\cdot56^{8}$ Cusp orbits $1^{4}\cdot2^{6}\cdot8^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $8$
$\Q$-gonality: $8 \le \gamma \le 16$
$\overline{\Q}$-gonality: $8 \le \gamma \le 16$
Rational cusps: $4$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.768.49.135

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}5&40\\12&1\end{bmatrix}$, $\begin{bmatrix}21&52\\12&21\end{bmatrix}$, $\begin{bmatrix}23&32\\6&53\end{bmatrix}$, $\begin{bmatrix}33&0\\50&51\end{bmatrix}$, $\begin{bmatrix}37&52\\4&5\end{bmatrix}$, $\begin{bmatrix}45&4\\6&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.1536.49-56.er.2.1, 56.1536.49-56.er.2.2, 56.1536.49-56.er.2.3, 56.1536.49-56.er.2.4, 56.1536.49-56.er.2.5, 56.1536.49-56.er.2.6, 56.1536.49-56.er.2.7, 56.1536.49-56.er.2.8, 56.1536.49-56.er.2.9, 56.1536.49-56.er.2.10, 56.1536.49-56.er.2.11, 56.1536.49-56.er.2.12, 56.1536.49-56.er.2.13, 56.1536.49-56.er.2.14, 56.1536.49-56.er.2.15, 56.1536.49-56.er.2.16, 56.1536.49-56.er.2.17, 56.1536.49-56.er.2.18, 56.1536.49-56.er.2.19, 56.1536.49-56.er.2.20, 56.1536.49-56.er.2.21, 56.1536.49-56.er.2.22, 56.1536.49-56.er.2.23, 56.1536.49-56.er.2.24, 56.1536.49-56.er.2.25, 56.1536.49-56.er.2.26, 56.1536.49-56.er.2.27, 56.1536.49-56.er.2.28, 56.1536.49-56.er.2.29, 56.1536.49-56.er.2.30, 56.1536.49-56.er.2.31, 56.1536.49-56.er.2.32
Cyclic 56-isogeny field degree: $2$
Cyclic 56-torsion field degree: $48$
Full 56-torsion field degree: $4032$

Jacobian

Conductor: $2^{208}\cdot7^{75}$
Simple: no
Squarefree: no
Decomposition: $1^{21}\cdot2^{6}\cdot4^{4}$
Newforms: 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a, 56.2.b.b, 112.2.a.a, 112.2.a.b, 112.2.a.c, 224.2.b.a, 224.2.b.b, 392.2.b.b, 392.2.b.c, 1568.2.b.a, 1568.2.b.d, 3136.2.a.bf, 3136.2.a.by, 3136.2.a.c, 3136.2.a.e, 3136.2.a.f, 3136.2.a.m$^{2}$, 3136.2.a.p, 3136.2.a.q, 3136.2.a.w, 3136.2.a.y, 3136.2.a.z

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.96.1.n.2 $56$ $8$ $8$ $1$ $1$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.384.23.f.2 $56$ $2$ $2$ $23$ $1$ $1^{10}\cdot2^{4}\cdot4^{2}$
56.384.23.g.1 $56$ $2$ $2$ $23$ $1$ $1^{10}\cdot2^{4}\cdot4^{2}$
56.384.25.bj.2 $56$ $2$ $2$ $25$ $8$ $2^{4}\cdot4^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.1536.97.dx.2 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.dx.3 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.dx.6 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.dx.7 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.eb.1 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.eb.4 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.eb.5 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.97.eb.8 $56$ $2$ $2$ $97$ $10$ $2^{12}\cdot4^{4}\cdot8$
56.1536.105.bv.1 $56$ $2$ $2$ $105$ $8$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.bv.2 $56$ $2$ $2$ $105$ $8$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.bx.3 $56$ $2$ $2$ $105$ $12$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.bx.4 $56$ $2$ $2$ $105$ $12$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.cy.2 $56$ $2$ $2$ $105$ $20$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.dd.2 $56$ $2$ $2$ $105$ $16$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.dl.1 $56$ $2$ $2$ $105$ $20$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.dr.3 $56$ $2$ $2$ $105$ $16$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.ff.2 $56$ $2$ $2$ $105$ $20$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.fl.2 $56$ $2$ $2$ $105$ $16$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.gg.1 $56$ $2$ $2$ $105$ $20$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.gl.3 $56$ $2$ $2$ $105$ $16$ $1^{20}\cdot2^{10}\cdot4^{4}$
56.1536.105.id.3 $56$ $2$ $2$ $105$ $8$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.id.4 $56$ $2$ $2$ $105$ $8$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.if.1 $56$ $2$ $2$ $105$ $12$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.1536.105.if.2 $56$ $2$ $2$ $105$ $12$ $2^{8}\cdot4^{2}\cdot8^{4}$
56.2304.145.lj.1 $56$ $3$ $3$ $145$ $10$ $2^{16}\cdot4^{4}\cdot12^{4}$
56.2304.145.lj.3 $56$ $3$ $3$ $145$ $10$ $2^{16}\cdot4^{4}\cdot12^{4}$
56.2304.145.sr.2 $56$ $3$ $3$ $145$ $30$ $1^{32}\cdot2^{8}\cdot6^{8}$
56.5376.385.il.1 $56$ $7$ $7$ $385$ $67$ $1^{86}\cdot2^{51}\cdot4^{13}\cdot6^{8}\cdot12^{4}$