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Properties

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

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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: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{2}\cdot4^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.1.544

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}13&18\\46&49\end{bmatrix}$, $\begin{bmatrix}27&44\\32&31\end{bmatrix}$, $\begin{bmatrix}43&2\\12&1\end{bmatrix}$, $\begin{bmatrix}53&22\\36&47\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.ce.1.1, 56.192.1-56.ce.1.2, 56.192.1-56.ce.1.3, 56.192.1-56.ce.1.4, 56.192.1-56.ce.1.5, 56.192.1-56.ce.1.6, 56.192.1-56.ce.1.7, 56.192.1-56.ce.1.8, 112.192.1-56.ce.1.1, 112.192.1-56.ce.1.2, 112.192.1-56.ce.1.3, 112.192.1-56.ce.1.4, 168.192.1-56.ce.1.1, 168.192.1-56.ce.1.2, 168.192.1-56.ce.1.3, 168.192.1-56.ce.1.4, 168.192.1-56.ce.1.5, 168.192.1-56.ce.1.6, 168.192.1-56.ce.1.7, 168.192.1-56.ce.1.8, 280.192.1-56.ce.1.1, 280.192.1-56.ce.1.2, 280.192.1-56.ce.1.3, 280.192.1-56.ce.1.4, 280.192.1-56.ce.1.5, 280.192.1-56.ce.1.6, 280.192.1-56.ce.1.7, 280.192.1-56.ce.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

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x^{2} + y^{2} + z^{2} $
$=$ $14 y^{2} - 14 z^{2} + w^{2}$
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Rational points

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

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{7^4}\cdot\frac{(38416z^{8}-5488z^{6}w^{2}+980z^{4}w^{4}-56z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(14z^{2}-w^{2})^{4}(28z^{2}-w^{2})^{2}}$

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.j.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.k.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.l.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.ba.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.bg.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bh.1 $56$ $2$ $2$ $1$ $1$ 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.of.1 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.blc.2 $56$ $21$ $21$ $145$ $24$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.bme.1 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
112.192.5.x.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.cf.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.ea.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.el.1 $112$ $2$ $2$ $5$ $?$ not computed
168.288.17.jdo.2 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.dsg.2 $168$ $4$ $4$ $17$ $?$ not computed