Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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: | 56.48.1.389 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}0&51\\23&4\end{bmatrix}$, $\begin{bmatrix}6&33\\21&22\end{bmatrix}$, $\begin{bmatrix}47&20\\36&37\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $64512$ |
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 $ | $=$ | $ 3 y^{2} - 2 y z - 2 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} + 16 x^{2} y^{2} + 21 x^{2} z^{2} + 18 y^{4} + 84 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^4}\cdot\frac{1915325720yz^{11}+4104269400yz^{9}w^{2}+2414676096yz^{7}w^{4}+89201952yz^{5}w^{6}-108718848yz^{3}w^{8}+16329600yzw^{10}+1050958517z^{12}+2976082718z^{10}w^{2}+2719103688z^{8}w^{4}+692006616z^{6}w^{6}-109438560z^{4}w^{8}-4626720z^{2}w^{10}-1944000w^{12}}{w^{4}(883568yz^{7}+1136016yz^{5}w^{2}+437472yz^{3}w^{4}+48384yzw^{6}+487403z^{8}+961772z^{6}w^{2}+602700z^{4}w^{4}+133056z^{2}w^{6}+6912w^{8})}$ |
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.v.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.0.bw.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.cf.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.cz.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dr.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.ba.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bv.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.mp.1 | $56$ | $8$ | $8$ | $25$ | $10$ | $1^{20}\cdot2^{2}$ |
56.1008.73.bnd.1 | $56$ | $21$ | $21$ | $73$ | $23$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.bmj.1 | $56$ | $28$ | $28$ | $97$ | $33$ | $1^{36}\cdot2^{28}\cdot4$ |
168.144.9.evx.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bno.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.yd.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.dex.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |