Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
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: | 10K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.199 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&7\\8&15\end{bmatrix}$, $\begin{bmatrix}21&16\\24&33\end{bmatrix}$, $\begin{bmatrix}31&23\\14&5\end{bmatrix}$, $\begin{bmatrix}39&38\\8&9\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $64$ |
Full 40-torsion field degree: | $10240$ |
Jacobian
Conductor: | $2^{4}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 3 x z - 2 y^{2} - 5 z^{2} - 2 w^{2} $ |
$=$ | $5 x^{2} - 5 x z - 2 y^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 10 x^{2} y^{2} - 4 x^{2} z^{2} + 4 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{108xz^{17}+864xz^{15}w^{2}-2880xz^{13}w^{4}-56576xz^{11}w^{6}-271360xz^{9}w^{8}-663552xz^{7}w^{10}-905216xz^{5}w^{12}-655360xz^{3}w^{14}-196608xzw^{16}-405z^{18}-6912z^{16}w^{2}-48240z^{14}w^{4}-177568z^{12}w^{6}-364544z^{10}w^{8}-384000z^{8}w^{10}-105472z^{6}w^{12}+163840z^{4}w^{14}+147456z^{2}w^{16}+32768w^{18}}{z^{10}(z^{2}+2w^{2})^{2}(4xz^{3}+16xzw^{2}-15z^{4}-46z^{2}w^{2}-16w^{4})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.36.1.d.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.co.1 | $40$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
40.36.0.a.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.e.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 |
---|---|---|---|---|---|---|
40.144.5.ew.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.fa.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.fy.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.gc.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.ih.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ik.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.jj.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.jm.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.360.13.cj.1 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
120.144.5.cmm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cmq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.coq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cou.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.edl.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.edo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.efp.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.efs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.tk.2 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.iea.2 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.be.1 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bdo.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bdq.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bec.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bee.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmf.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmg.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmt.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmu.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |