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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.386 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}10&43\\17&2\end{bmatrix}$, $\begin{bmatrix}39&20\\20&29\end{bmatrix}$, $\begin{bmatrix}40&9\\25&48\end{bmatrix}$ |
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 $ | $=$ | $ 2 y^{2} + y z + z^{2} + w^{2} $ |
$=$ | $56 x^{2} - 3 y^{2} + 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 3 x^{2} y^{2} + 112 x^{2} z^{2} + 2 y^{4} + 84 y^{2} z^{2} + 882 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{5294205yz^{11}+15126300yz^{9}w^{2}+20398896yz^{7}w^{4}+15212736yz^{5}w^{6}-17054352yz^{3}w^{8}-12196800yzw^{10}+2705927z^{12}+8521149z^{10}w^{2}-1310946z^{8}w^{4}-21167216z^{6}w^{6}-25565064z^{4}w^{8}-9738960z^{2}w^{10}+5324000w^{12}}{w^{4}(7203yz^{7}+12348yz^{5}w^{2}+15288yz^{3}w^{4}+6048yzw^{6}-16807z^{8}-39445z^{6}w^{2}-25774z^{4}w^{4}-4536z^{2}w^{6}+648w^{8})}$ |
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.bf.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.0.cp.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.cv.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dr.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dy.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.be.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bu.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.ot.1 | $56$ | $8$ | $8$ | $25$ | $7$ | $1^{20}\cdot2^{2}$ |
56.1008.73.brn.1 | $56$ | $21$ | $21$ | $73$ | $35$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.bqt.1 | $56$ | $28$ | $28$ | $97$ | $42$ | $1^{36}\cdot2^{28}\cdot4$ |
112.96.3.lo.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.lq.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.pg.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.pi.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.144.9.fff.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bsg.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.bcn.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.dlt.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |