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Properties

Label 56.96.1.n.2
Level $56$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $2$

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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 $2$ are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $1^{2}\cdot2^{3}\cdot8$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.1.205

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}1&52\\8&41\end{bmatrix}$, $\begin{bmatrix}25&52\\48&39\end{bmatrix}$, $\begin{bmatrix}33&36\\36&29\end{bmatrix}$, $\begin{bmatrix}35&4\\20&37\end{bmatrix}$, $\begin{bmatrix}45&8\\32&39\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.n.2.1, 56.192.1-56.n.2.2, 56.192.1-56.n.2.3, 56.192.1-56.n.2.4, 56.192.1-56.n.2.5, 56.192.1-56.n.2.6, 56.192.1-56.n.2.7, 56.192.1-56.n.2.8, 56.192.1-56.n.2.9, 56.192.1-56.n.2.10, 56.192.1-56.n.2.11, 56.192.1-56.n.2.12, 56.192.1-56.n.2.13, 56.192.1-56.n.2.14, 56.192.1-56.n.2.15, 56.192.1-56.n.2.16, 168.192.1-56.n.2.1, 168.192.1-56.n.2.2, 168.192.1-56.n.2.3, 168.192.1-56.n.2.4, 168.192.1-56.n.2.5, 168.192.1-56.n.2.6, 168.192.1-56.n.2.7, 168.192.1-56.n.2.8, 168.192.1-56.n.2.9, 168.192.1-56.n.2.10, 168.192.1-56.n.2.11, 168.192.1-56.n.2.12, 168.192.1-56.n.2.13, 168.192.1-56.n.2.14, 168.192.1-56.n.2.15, 168.192.1-56.n.2.16, 280.192.1-56.n.2.1, 280.192.1-56.n.2.2, 280.192.1-56.n.2.3, 280.192.1-56.n.2.4, 280.192.1-56.n.2.5, 280.192.1-56.n.2.6, 280.192.1-56.n.2.7, 280.192.1-56.n.2.8, 280.192.1-56.n.2.9, 280.192.1-56.n.2.10, 280.192.1-56.n.2.11, 280.192.1-56.n.2.12, 280.192.1-56.n.2.13, 280.192.1-56.n.2.14, 280.192.1-56.n.2.15, 280.192.1-56.n.2.16
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
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.b.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.b.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.n.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
56.192.5.d.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.f.3 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.h.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.k.3 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.p.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.s.4 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.z.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.bb.4 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.768.49.er.2 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.nd.1 $56$ $21$ $21$ $145$ $24$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.nx.2 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
168.192.5.bu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bw.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.do.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dq.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.eu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ew.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ha.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.hc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.288.17.bpg.1 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.po.1 $168$ $4$ $4$ $17$ $?$ not computed
280.192.5.bm.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.bo.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.dg.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.di.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.em.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.eo.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.gs.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.gu.2 $280$ $2$ $2$ $5$ $?$ not computed