Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
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 | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | 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: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.164 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&19\\38&25\end{bmatrix}$, $\begin{bmatrix}23&18\\14&17\end{bmatrix}$, $\begin{bmatrix}25&7\\36&1\end{bmatrix}$, $\begin{bmatrix}33&5\\16&37\end{bmatrix}$, $\begin{bmatrix}39&18\\20&27\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.144.1-40.m.2.1, 40.144.1-40.m.2.2, 40.144.1-40.m.2.3, 40.144.1-40.m.2.4, 40.144.1-40.m.2.5, 40.144.1-40.m.2.6, 40.144.1-40.m.2.7, 40.144.1-40.m.2.8, 120.144.1-40.m.2.1, 120.144.1-40.m.2.2, 120.144.1-40.m.2.3, 120.144.1-40.m.2.4, 120.144.1-40.m.2.5, 120.144.1-40.m.2.6, 120.144.1-40.m.2.7, 120.144.1-40.m.2.8, 280.144.1-40.m.2.1, 280.144.1-40.m.2.2, 280.144.1-40.m.2.3, 280.144.1-40.m.2.4, 280.144.1-40.m.2.5, 280.144.1-40.m.2.6, 280.144.1-40.m.2.7, 280.144.1-40.m.2.8 |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $64$ |
Full 40-torsion field degree: | $10240$ |
Jacobian
Conductor: | $2^{4}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + y z $ |
$=$ | $y^{2} + 2 y z + 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{2} z^{2} + 2 y^{2} z^{2} + 20 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{216yz^{17}+432yz^{15}w^{2}-360yz^{13}w^{4}-1768yz^{11}w^{6}-2120yz^{9}w^{8}-1296yz^{7}w^{10}-442yz^{5}w^{12}-80yz^{3}w^{14}-6yzw^{16}-2160z^{18}-11664z^{16}w^{2}-25920z^{14}w^{4}-31036z^{12}w^{6}-21992z^{10}w^{8}-9480z^{8}w^{10}-2416z^{6}w^{12}-320z^{4}w^{14}-12z^{2}w^{16}+w^{18}}{z^{10}(2z^{2}+w^{2})^{2}(2yz^{3}+2yzw^{2}-20z^{4}-13z^{2}w^{2}-2w^{4})}$ |
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.b.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.36.0.a.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.d.2 | $40$ | $2$ | $2$ | $0$ | $0$ | 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.b.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.bd.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.ce.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.ci.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.ei.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.en.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ew.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.ey.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.360.13.y.1 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
120.144.5.bca.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bcc.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bco.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bcq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.boa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.boc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.boo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.boq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.ds.2 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.eoa.2 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.m.2 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.pd.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.pe.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.pk.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.pl.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.rh.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.ri.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.ro.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.rp.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |