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Properties

Label 56.252.16.ci.1
Level $56$
Index $252$
Genus $16$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $252$ $\PSL_2$-index:$252$
Genus: $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $7^{6}\cdot14^{3}\cdot56^{3}$ Cusp orbits $3^{2}\cdot6$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $5 \le \gamma \le 8$
$\overline{\Q}$-gonality: $5 \le \gamma \le 8$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 56B16
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.252.16.6

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}4&15\\25&10\end{bmatrix}$, $\begin{bmatrix}17&46\\4&15\end{bmatrix}$, $\begin{bmatrix}34&5\\41&50\end{bmatrix}$, $\begin{bmatrix}42&25\\53&26\end{bmatrix}$, $\begin{bmatrix}49&36\\20&21\end{bmatrix}$, $\begin{bmatrix}51&54\\26&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.504.16-56.ci.1.1, 56.504.16-56.ci.1.2, 56.504.16-56.ci.1.3, 56.504.16-56.ci.1.4, 56.504.16-56.ci.1.5, 56.504.16-56.ci.1.6, 56.504.16-56.ci.1.7, 56.504.16-56.ci.1.8, 56.504.16-56.ci.1.9, 56.504.16-56.ci.1.10, 56.504.16-56.ci.1.11, 56.504.16-56.ci.1.12, 56.504.16-56.ci.1.13, 56.504.16-56.ci.1.14, 56.504.16-56.ci.1.15, 56.504.16-56.ci.1.16, 56.504.16-56.ci.1.17, 56.504.16-56.ci.1.18, 56.504.16-56.ci.1.19, 56.504.16-56.ci.1.20, 56.504.16-56.ci.1.21, 56.504.16-56.ci.1.22, 56.504.16-56.ci.1.23, 56.504.16-56.ci.1.24, 56.504.16-56.ci.1.25, 56.504.16-56.ci.1.26, 56.504.16-56.ci.1.27, 56.504.16-56.ci.1.28, 56.504.16-56.ci.1.29, 56.504.16-56.ci.1.30, 56.504.16-56.ci.1.31, 56.504.16-56.ci.1.32, 168.504.16-56.ci.1.1, 168.504.16-56.ci.1.2, 168.504.16-56.ci.1.3, 168.504.16-56.ci.1.4, 168.504.16-56.ci.1.5, 168.504.16-56.ci.1.6, 168.504.16-56.ci.1.7, 168.504.16-56.ci.1.8, 168.504.16-56.ci.1.9, 168.504.16-56.ci.1.10, 168.504.16-56.ci.1.11, 168.504.16-56.ci.1.12, 168.504.16-56.ci.1.13, 168.504.16-56.ci.1.14, 168.504.16-56.ci.1.15, 168.504.16-56.ci.1.16, 168.504.16-56.ci.1.17, 168.504.16-56.ci.1.18, 168.504.16-56.ci.1.19, 168.504.16-56.ci.1.20, 168.504.16-56.ci.1.21, 168.504.16-56.ci.1.22, 168.504.16-56.ci.1.23, 168.504.16-56.ci.1.24, 168.504.16-56.ci.1.25, 168.504.16-56.ci.1.26, 168.504.16-56.ci.1.27, 168.504.16-56.ci.1.28, 168.504.16-56.ci.1.29, 168.504.16-56.ci.1.30, 168.504.16-56.ci.1.31, 168.504.16-56.ci.1.32, 280.504.16-56.ci.1.1, 280.504.16-56.ci.1.2, 280.504.16-56.ci.1.3, 280.504.16-56.ci.1.4, 280.504.16-56.ci.1.5, 280.504.16-56.ci.1.6, 280.504.16-56.ci.1.7, 280.504.16-56.ci.1.8, 280.504.16-56.ci.1.9, 280.504.16-56.ci.1.10, 280.504.16-56.ci.1.11, 280.504.16-56.ci.1.12, 280.504.16-56.ci.1.13, 280.504.16-56.ci.1.14, 280.504.16-56.ci.1.15, 280.504.16-56.ci.1.16, 280.504.16-56.ci.1.17, 280.504.16-56.ci.1.18, 280.504.16-56.ci.1.19, 280.504.16-56.ci.1.20, 280.504.16-56.ci.1.21, 280.504.16-56.ci.1.22, 280.504.16-56.ci.1.23, 280.504.16-56.ci.1.24, 280.504.16-56.ci.1.25, 280.504.16-56.ci.1.26, 280.504.16-56.ci.1.27, 280.504.16-56.ci.1.28, 280.504.16-56.ci.1.29, 280.504.16-56.ci.1.30, 280.504.16-56.ci.1.31, 280.504.16-56.ci.1.32
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $12288$

Jacobian

Conductor: $2^{64}\cdot7^{32}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{6}$
Newforms: 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 3136.2.a.bc, 3136.2.a.bm, 3136.2.a.bp, 3136.2.a.bs, 3136.2.a.j, 3136.2.a.s

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
$X_{\mathrm{ns}}^+(7)$ $7$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.m.1 $8$ $21$ $21$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.m.1 $8$ $21$ $21$ $0$ $0$ full Jacobian
28.126.7.c.1 $28$ $2$ $2$ $7$ $0$ $1^{3}\cdot2^{3}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.504.31.nw.1 $56$ $2$ $2$ $31$ $10$ $1^{13}\cdot2$
56.504.31.ny.1 $56$ $2$ $2$ $31$ $9$ $1^{13}\cdot2$
56.504.31.oa.1 $56$ $2$ $2$ $31$ $6$ $1^{13}\cdot2$
56.504.31.oc.1 $56$ $2$ $2$ $31$ $3$ $1^{13}\cdot2$
56.504.31.pc.1 $56$ $2$ $2$ $31$ $4$ $1^{13}\cdot2$
56.504.31.pe.1 $56$ $2$ $2$ $31$ $4$ $1^{13}\cdot2$
56.504.31.pg.1 $56$ $2$ $2$ $31$ $7$ $1^{13}\cdot2$
56.504.31.pi.1 $56$ $2$ $2$ $31$ $5$ $1^{13}\cdot2$
56.504.34.cg.1 $56$ $2$ $2$ $34$ $10$ $1^{6}\cdot2^{6}$
56.504.34.ci.1 $56$ $2$ $2$ $34$ $4$ $1^{6}\cdot2^{6}$
56.504.34.dg.1 $56$ $2$ $2$ $34$ $4$ $1^{6}\cdot2^{6}$
56.504.34.di.1 $56$ $2$ $2$ $34$ $10$ $1^{6}\cdot2^{6}$
56.504.34.ey.1 $56$ $2$ $2$ $34$ $8$ $1^{6}\cdot2^{6}$
56.504.34.fa.1 $56$ $2$ $2$ $34$ $6$ $1^{6}\cdot2^{6}$
56.504.34.fc.1 $56$ $2$ $2$ $34$ $6$ $1^{6}\cdot2^{6}$
56.504.34.fe.1 $56$ $2$ $2$ $34$ $8$ $1^{6}\cdot2^{6}$
56.504.34.gm.1 $56$ $2$ $2$ $34$ $5$ $1^{14}\cdot2^{2}$
56.504.34.go.1 $56$ $2$ $2$ $34$ $8$ $1^{14}\cdot2^{2}$
56.504.34.gq.1 $56$ $2$ $2$ $34$ $8$ $1^{14}\cdot2^{2}$
56.504.34.gs.1 $56$ $2$ $2$ $34$ $5$ $1^{14}\cdot2^{2}$
56.504.34.hs.1 $56$ $2$ $2$ $34$ $3$ $1^{14}\cdot2^{2}$
56.504.34.hu.1 $56$ $2$ $2$ $34$ $12$ $1^{14}\cdot2^{2}$
56.504.34.hw.1 $56$ $2$ $2$ $34$ $12$ $1^{14}\cdot2^{2}$
56.504.34.hy.1 $56$ $2$ $2$ $34$ $3$ $1^{14}\cdot2^{2}$