Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
Index: | $192$ | $\PSL_2$-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 | $2^{8}$ | ||
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: | 56.192.1.922 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&28\\12&13\end{bmatrix}$, $\begin{bmatrix}11&20\\20&43\end{bmatrix}$, $\begin{bmatrix}37&24\\0&41\end{bmatrix}$, $\begin{bmatrix}47&20\\14&45\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.96.1.f.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $192$ |
Full 56-torsion field degree: | $16128$ |
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 $ | $=$ | $ x^{2} - 2 x w - y^{2} $ |
$=$ | $y^{2} + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 6 x^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{8}-2z^{6}w^{2}+5z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(z-w)^{4}(z+w)^{4}(2z^{2}-w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.1.f.1 :
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}-2X^{2}Y^{2}-6X^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.96.0-8.b.2.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.b.2.4 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.c.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.c.1.9 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.h.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.h.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.i.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.0-8.i.1.7 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.96.1-8.g.1.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.96.1-8.g.1.4 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.96.1-8.m.2.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.96.1-8.m.2.6 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.96.1-8.n.1.2 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.96.1-8.n.1.7 | $56$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
56.384.5-8.c.1.2 | $56$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
56.384.5-8.d.3.2 | $56$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
56.384.5-56.z.1.6 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
56.384.5-56.ba.1.4 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
56.1536.49-56.fr.2.11 | $56$ | $8$ | $8$ | $49$ | $6$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
56.4032.145-56.pd.2.5 | $56$ | $21$ | $21$ | $145$ | $23$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
56.5376.193-56.px.2.6 | $56$ | $28$ | $28$ | $193$ | $29$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
112.384.5-16.b.1.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-16.i.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.j.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-16.o.1.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-16.v.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.bl.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.bs.1.4 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.cx.2.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-24.bh.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-24.bi.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.hp.2.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.hr.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.384.5-40.z.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.384.5-40.ba.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.384.5-280.hh.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.384.5-280.hj.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |