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Properties

Label 56.96.1.p.2
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: $64$
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: $0$
$\Q$-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: 56.96.1.761

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}3&0\\0&43\end{bmatrix}$, $\begin{bmatrix}19&16\\50&33\end{bmatrix}$, $\begin{bmatrix}37&8\\22&47\end{bmatrix}$, $\begin{bmatrix}37&48\\24&25\end{bmatrix}$, $\begin{bmatrix}43&12\\38&37\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.p.2.1, 56.192.1-56.p.2.2, 56.192.1-56.p.2.3, 56.192.1-56.p.2.4, 56.192.1-56.p.2.5, 56.192.1-56.p.2.6, 56.192.1-56.p.2.7, 56.192.1-56.p.2.8, 56.192.1-56.p.2.9, 56.192.1-56.p.2.10, 56.192.1-56.p.2.11, 56.192.1-56.p.2.12, 168.192.1-56.p.2.1, 168.192.1-56.p.2.2, 168.192.1-56.p.2.3, 168.192.1-56.p.2.4, 168.192.1-56.p.2.5, 168.192.1-56.p.2.6, 168.192.1-56.p.2.7, 168.192.1-56.p.2.8, 168.192.1-56.p.2.9, 168.192.1-56.p.2.10, 168.192.1-56.p.2.11, 168.192.1-56.p.2.12, 280.192.1-56.p.2.1, 280.192.1-56.p.2.2, 280.192.1-56.p.2.3, 280.192.1-56.p.2.4, 280.192.1-56.p.2.5, 280.192.1-56.p.2.6, 280.192.1-56.p.2.7, 280.192.1-56.p.2.8, 280.192.1-56.p.2.9, 280.192.1-56.p.2.10, 280.192.1-56.p.2.11, 280.192.1-56.p.2.12
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $32256$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 7 x^{2} + 7 y^{2} + w^{2} $
$=$ $7 x^{2} - 7 y^{2} + 2 z^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{(4z^{4}-4z^{3}w+2z^{2}w^{2}-2zw^{3}+w^{4})^{3}(4z^{4}+4z^{3}w+2z^{2}w^{2}+2zw^{3}+w^{4})^{3}}{w^{8}z^{8}(2z^{2}-w^{2})^{2}(2z^{2}+w^{2})^{2}}$

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
8.48.1.g.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.x.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.y.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.x.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.y.1 $56$ $2$ $2$ $1$ $0$ 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.o.1 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.192.5.p.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.q.1 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.r.1 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.768.49.fg.2 $56$ $8$ $8$ $49$ $6$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.oo.1 $56$ $21$ $21$ $145$ $23$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.pi.1 $56$ $28$ $28$ $193$ $29$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
168.192.5.fp.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fq.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fr.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fs.1 $168$ $2$ $2$ $5$ $?$ not computed
168.288.17.buj.1 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.rz.1 $168$ $4$ $4$ $17$ $?$ not computed
280.192.5.fh.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fi.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fj.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fk.1 $280$ $2$ $2$ $5$ $?$ not computed