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Properties

Label 56.96.1.m.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: $32$
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: 56.96.1.777

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}25&8\\12&47\end{bmatrix}$, $\begin{bmatrix}25&52\\4&9\end{bmatrix}$, $\begin{bmatrix}27&28\\16&55\end{bmatrix}$, $\begin{bmatrix}31&52\\28&39\end{bmatrix}$, $\begin{bmatrix}47&52\\28&37\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.m.2.1, 56.192.1-56.m.2.2, 56.192.1-56.m.2.3, 56.192.1-56.m.2.4, 56.192.1-56.m.2.5, 56.192.1-56.m.2.6, 56.192.1-56.m.2.7, 56.192.1-56.m.2.8, 168.192.1-56.m.2.1, 168.192.1-56.m.2.2, 168.192.1-56.m.2.3, 168.192.1-56.m.2.4, 168.192.1-56.m.2.5, 168.192.1-56.m.2.6, 168.192.1-56.m.2.7, 168.192.1-56.m.2.8, 280.192.1-56.m.2.1, 280.192.1-56.m.2.2, 280.192.1-56.m.2.3, 280.192.1-56.m.2.4, 280.192.1-56.m.2.5, 280.192.1-56.m.2.6, 280.192.1-56.m.2.7, 280.192.1-56.m.2.8
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $32256$

Jacobian

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

Models

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

$ 0 $ $=$ $ 7 x y + 7 y^{2} + z w $
$=$ $7 x^{2} + 7 x y - 7 y^{2} + z^{2} - z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 7 x^{4} - 49 x^{2} y^{2} - x^{2} z^{2} - 7 y^{2} z^{2} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{7}z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

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

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.h.2 $8$ $2$ $2$ $1$ $0$ dimension zero
56.48.0.b.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.bo.1 $56$ $2$ $2$ $0$ $0$ full Jacobian

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.c.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.j.3 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.192.5.r.3 $56$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.192.5.y.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.768.49.en.2 $56$ $8$ $8$ $49$ $3$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.mx.2 $56$ $21$ $21$ $145$ $19$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.nr.1 $56$ $28$ $28$ $193$ $22$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
168.192.5.bj.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dn.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.et.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.gp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.288.17.bof.1 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.ox.2 $168$ $4$ $4$ $17$ $?$ not computed
280.192.5.bb.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.df.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.el.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.gh.2 $280$ $2$ $2$ $5$ $?$ not computed