Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
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 | $2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.204 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&5\\37&4\end{bmatrix}$, $\begin{bmatrix}12&9\\13&34\end{bmatrix}$, $\begin{bmatrix}30&35\\9&10\end{bmatrix}$ |
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: | $2^{6}\cdot7^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} - x y + x z + y^{2} - 2 y z + 2 z^{2} $ |
$=$ | $x^{2} - 4 x y - 4 x z + 4 y^{2} + 8 y z + 8 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 680 x^{4} + 1232 x^{3} y + 896 x^{2} y^{2} - 65 x^{2} z^{2} + 392 x y^{3} - 70 x y z^{2} + 98 y^{4} + \cdots + 2 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{1728xz^{9}w^{2}-2304xz^{7}w^{4}+224xz^{5}w^{6}+320xz^{3}w^{8}+12xzw^{10}-3456yz^{9}w^{2}+4608yz^{7}w^{4}-448yz^{5}w^{6}-640yz^{3}w^{8}-24yzw^{10}-1728z^{12}+5040z^{8}w^{4}-3200z^{6}w^{6}+92z^{4}w^{8}+96z^{2}w^{10}+w^{12}}{z^{8}(4xzw^{2}-8yzw^{2}-16z^{4}+w^{4})}$ |
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 |
---|---|---|---|---|---|---|
8.24.0.bl.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.ef.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.dh.1 | $56$ | $2$ | $2$ | $1$ | $1$ | 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.qm.2 | $56$ | $8$ | $8$ | $25$ | $4$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
56.1008.73.eav.2 | $56$ | $21$ | $21$ | $73$ | $18$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
56.1344.97.dvb.1 | $56$ | $28$ | $28$ | $97$ | $21$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
112.96.5.hq.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.hs.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.jm.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.jo.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.rm.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ro.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ti.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.tk.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.9.bavx.2 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.dil.2 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.efp.2 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.qnz.2 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |