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Properties

Label 56.96.1.ch.1
Level $56$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $3136$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $1^{4}\cdot4^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.1.560

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}17&52\\24&53\end{bmatrix}$, $\begin{bmatrix}21&26\\54&41\end{bmatrix}$, $\begin{bmatrix}53&26\\4&11\end{bmatrix}$, $\begin{bmatrix}55&38\\24&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.ch.1.1, 56.192.1-56.ch.1.2, 56.192.1-56.ch.1.3, 56.192.1-56.ch.1.4, 56.192.1-56.ch.1.5, 56.192.1-56.ch.1.6, 56.192.1-56.ch.1.7, 56.192.1-56.ch.1.8, 112.192.1-56.ch.1.1, 112.192.1-56.ch.1.2, 112.192.1-56.ch.1.3, 112.192.1-56.ch.1.4, 112.192.1-56.ch.1.5, 112.192.1-56.ch.1.6, 112.192.1-56.ch.1.7, 112.192.1-56.ch.1.8, 112.192.1-56.ch.1.9, 112.192.1-56.ch.1.10, 112.192.1-56.ch.1.11, 112.192.1-56.ch.1.12, 168.192.1-56.ch.1.1, 168.192.1-56.ch.1.2, 168.192.1-56.ch.1.3, 168.192.1-56.ch.1.4, 168.192.1-56.ch.1.5, 168.192.1-56.ch.1.6, 168.192.1-56.ch.1.7, 168.192.1-56.ch.1.8, 280.192.1-56.ch.1.1, 280.192.1-56.ch.1.2, 280.192.1-56.ch.1.3, 280.192.1-56.ch.1.4, 280.192.1-56.ch.1.5, 280.192.1-56.ch.1.6, 280.192.1-56.ch.1.7, 280.192.1-56.ch.1.8
Cyclic 56-isogeny field degree: $8$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $32256$

Jacobian

Conductor: $2^{6}\cdot7^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 3136.2.a.m

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.l.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.m.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.n.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.ba.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.bi.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bj.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bv.1 $56$ $2$ $2$ $1$ $1$ 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
56.768.49.oi.2 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.blj.2 $56$ $21$ $21$ $145$ $24$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.bml.1 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
112.192.5.br.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.cl.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dl.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dm.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dp.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dq.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dt.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.du.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dx.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.dy.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.eg.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.eo.1 $112$ $2$ $2$ $5$ $?$ not computed
168.288.17.jed.2 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.dsn.2 $168$ $4$ $4$ $17$ $?$ not computed