Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
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: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.94 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&4\\1&25\end{bmatrix}$, $\begin{bmatrix}9&16\\5&11\end{bmatrix}$, $\begin{bmatrix}13&0\\17&19\end{bmatrix}$, $\begin{bmatrix}29&22\\6&7\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $30720$ |
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 $ | $=$ | $ 10 x y - z w $ |
$=$ | $80 x^{2} + 10 y^{2} + 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 10 x^{4} + 25 x^{2} y^{2} + x^{2} z^{2} + 5 y^{2} z^{2} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{5600y^{2}z^{4}-10800y^{2}z^{2}w^{2}+4850y^{2}w^{4}+1728z^{6}-4496z^{4}w^{2}+3124z^{2}w^{4}-27w^{6}}{160y^{2}z^{4}+80y^{2}z^{2}w^{2}-10y^{2}w^{4}+64z^{6}+16z^{4}w^{2}+12z^{2}w^{4}-w^{6}}$ |
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 |
---|---|---|---|---|---|---|
8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.12.0.bq.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.0.bt.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.48.1.i.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.cs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.dh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.du.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fy.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gp.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.120.9.cj.1 | $40$ | $5$ | $5$ | $9$ | $5$ | $1^{6}\cdot2$ |
40.144.9.dx.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
40.240.17.nv.1 | $40$ | $10$ | $10$ | $17$ | $9$ | $1^{12}\cdot2^{2}$ |
120.48.1.ph.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.pl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.qn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.qr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.72.5.fd.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.dd.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.qx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.rb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.rn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.rr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.up.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ut.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.vf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.vj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.192.13.dd.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |