Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.392 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}24&5\\5&44\end{bmatrix}$, $\begin{bmatrix}35&32\\40&17\end{bmatrix}$, $\begin{bmatrix}50&9\\43&50\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 112.96.1-56.gf.1.1, 112.96.1-56.gf.1.2, 112.96.1-56.gf.1.3, 112.96.1-56.gf.1.4 |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $64512$ |
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 $ | $=$ | $ 9 y^{2} + 8 y z + 8 z^{2} + 2 w^{2} $ |
$=$ | $28 x^{2} + 2 y^{2} + y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 20 x^{2} y^{2} + 21 x^{2} z^{2} + 162 y^{4} - 252 y^{2} z^{2} + 98 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{3196523330yz^{11}-15868698804yz^{9}w^{2}-1673314524yz^{7}w^{4}+2632494816yz^{5}w^{6}+782504226yz^{3}w^{8}+124002900yzw^{10}-4997847169z^{12}-10191260590z^{10}w^{2}+7314624891z^{8}w^{4}+4480064316z^{6}w^{6}+354102273z^{4}w^{8}+17911530z^{2}w^{10}+7381125w^{12}}{12786093320yz^{11}+62952635340yz^{9}w^{2}+56675653020yz^{7}w^{4}+16417085832yz^{5}w^{6}+2643925536yz^{3}w^{8}+317447424yzw^{10}-19991388676z^{12}-5494746124z^{10}w^{2}+25725752199z^{8}w^{4}+17209123596z^{6}w^{6}+3954886236z^{4}w^{8}+379173312z^{2}w^{10}-15116544w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.1.z.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.0.cj.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.cm.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dh.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dx.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.z.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bh.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 |
---|---|---|---|---|---|---|
56.384.25.nx.1 | $56$ | $8$ | $8$ | $25$ | $11$ | $1^{20}\cdot2^{2}$ |
56.1008.73.bpl.1 | $56$ | $21$ | $21$ | $73$ | $22$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.bor.1 | $56$ | $28$ | $28$ | $97$ | $33$ | $1^{36}\cdot2^{28}\cdot4$ |
168.144.9.fbh.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bqu.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.bbb.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.djr.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |