Properties

Label 56.96.1.q.1
Level $56$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $32$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.1.745

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}11&48\\10&41\end{bmatrix}$, $\begin{bmatrix}15&16\\0&23\end{bmatrix}$, $\begin{bmatrix}27&12\\30&49\end{bmatrix}$, $\begin{bmatrix}35&44\\24&39\end{bmatrix}$, $\begin{bmatrix}53&8\\18&27\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.q.1.1, 56.192.1-56.q.1.2, 56.192.1-56.q.1.3, 56.192.1-56.q.1.4, 56.192.1-56.q.1.5, 56.192.1-56.q.1.6, 56.192.1-56.q.1.7, 56.192.1-56.q.1.8, 56.192.1-56.q.1.9, 56.192.1-56.q.1.10, 56.192.1-56.q.1.11, 56.192.1-56.q.1.12, 112.192.1-56.q.1.1, 112.192.1-56.q.1.2, 112.192.1-56.q.1.3, 112.192.1-56.q.1.4, 112.192.1-56.q.1.5, 112.192.1-56.q.1.6, 112.192.1-56.q.1.7, 112.192.1-56.q.1.8, 168.192.1-56.q.1.1, 168.192.1-56.q.1.2, 168.192.1-56.q.1.3, 168.192.1-56.q.1.4, 168.192.1-56.q.1.5, 168.192.1-56.q.1.6, 168.192.1-56.q.1.7, 168.192.1-56.q.1.8, 168.192.1-56.q.1.9, 168.192.1-56.q.1.10, 168.192.1-56.q.1.11, 168.192.1-56.q.1.12, 280.192.1-56.q.1.1, 280.192.1-56.q.1.2, 280.192.1-56.q.1.3, 280.192.1-56.q.1.4, 280.192.1-56.q.1.5, 280.192.1-56.q.1.6, 280.192.1-56.q.1.7, 280.192.1-56.q.1.8, 280.192.1-56.q.1.9, 280.192.1-56.q.1.10, 280.192.1-56.q.1.11, 280.192.1-56.q.1.12
Cyclic 56-isogeny field degree: $8$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $32256$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Rational points

This modular curve has no real points, and therefore no rational points.

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.1.h.1 $8$ $2$ $2$ $1$ $0$ dimension zero
56.48.0.b.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.c.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.ba.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.bb.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.q.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.s.2 $56$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.192.5.r.1 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.192.5.r.2 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.192.5.s.1 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.s.4 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.768.49.fh.1 $56$ $8$ $8$ $49$ $3$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.op.2 $56$ $21$ $21$ $145$ $19$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.pj.2 $56$ $28$ $28$ $193$ $22$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
112.192.5.bg.3 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bg.4 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bh.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bh.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.9.fg.3 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.fg.4 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.fh.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.fh.2 $112$ $2$ $2$ $9$ $?$ not computed
168.192.5.ft.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ft.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fu.3 $168$ $2$ $2$ $5$ $?$ not computed
168.288.17.buk.2 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.sa.2 $168$ $4$ $4$ $17$ $?$ not computed
280.192.5.fl.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fl.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fm.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fm.2 $280$ $2$ $2$ $5$ $?$ not computed