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: | 40.24.1.88 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&28\\25&33\end{bmatrix}$, $\begin{bmatrix}9&38\\35&7\end{bmatrix}$, $\begin{bmatrix}23&10\\14&13\end{bmatrix}$, $\begin{bmatrix}27&22\\36&17\end{bmatrix}$ |
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 $ | $=$ | $ x^{2} + 5 x y + x z + x w + z w $ |
$=$ | $43 x^{2} - 10 x y + 6 x z + 5 y^{2} + 4 z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 226 x^{4} + 42 x^{3} y + 12 x^{3} z + 21 x^{2} y^{2} + 4 x^{2} y z + 6 x^{2} z^{2} + 2 x y^{2} z + \cdots + 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3}{5^3}\cdot\frac{110075514264xz^{5}+6517772820xz^{4}w-335753769800xz^{3}w^{2}+124224838020xz^{2}w^{3}+105168582960xzw^{4}-4287593264xw^{5}+12626357265y^{2}z^{4}+57684684240y^{2}z^{3}w-72139638860y^{2}z^{2}w^{2}-47462187760y^{2}zw^{3}+40298930740y^{2}w^{4}-35373903000yz^{4}w+24019492500yz^{3}w^{2}+48014527500yz^{2}w^{3}-31908237000yzw^{4}+318244152z^{6}+53222527992z^{5}w-58500926620z^{4}w^{2}-56294640920z^{3}w^{3}+63663788560z^{2}w^{4}-3110790152zw^{5}-1723055512w^{6}}{18505200xz^{5}+1457060xz^{4}w+6387240xz^{3}w^{2}+4292500xz^{2}w^{3}+1858086xzw^{4}-815076xw^{5}+154875y^{2}z^{4}-1608400y^{2}z^{3}w+2408500y^{2}z^{2}w^{2}-1134000y^{2}zw^{3}-477630y^{2}w^{4}-4347000yz^{4}w-1183500yz^{3}w^{2}-1282500yz^{2}w^{3}+520020yzw^{4}+123900z^{6}-417320z^{5}w+1258700z^{4}w^{2}-125480z^{3}w^{3}+1176545z^{2}w^{4}-330804zw^{5}-95526w^{6}}$ |
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 |
20.12.0.n.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.0.bu.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.e.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.cr.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.dx.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.ek.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fe.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fg.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.hb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.hh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.120.9.cq.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
40.144.9.eu.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
40.240.17.pi.1 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
120.48.1.pw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.qa.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.we.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.xz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.yf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.72.5.hg.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.ea.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.sc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.sg.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ss.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.sw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.vu.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.vy.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.xa.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.xe.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.192.13.ea.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |