Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
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: | $1$ | ||||||
$\Q$-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: | 40.72.1.116 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&21\\36&25\end{bmatrix}$, $\begin{bmatrix}5&36\\2&19\end{bmatrix}$, $\begin{bmatrix}23&23\\34&37\end{bmatrix}$, $\begin{bmatrix}25&38\\18&15\end{bmatrix}$ |
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: | $2^{6}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1600.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x z + w^{2} $ |
$=$ | $x^{2} + 5 y^{2} + 5 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} + 5 x^{2} y^{2} + 4 x^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{62500000y^{18}-75000000y^{16}w^{2}+15000000y^{14}w^{4}+17000000y^{12}w^{6}-10650000y^{10}w^{8}+2220000y^{8}w^{10}-149000y^{6}w^{12}+46800y^{4}w^{14}-50985y^{2}w^{16}+62496000z^{18}+187488000z^{16}w^{2}+253080000z^{14}w^{4}+204912000z^{12}w^{6}+111960000z^{10}w^{8}+43956000z^{8}w^{10}+12776400z^{6}w^{12}+2719800z^{4}w^{14}+370350z^{2}w^{16}+32382w^{18}}{w^{4}z^{2}(2z^{2}+w^{2})^{5}(10z^{2}+w^{2})}$ |
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 |
---|---|---|---|---|---|---|
20.36.0.a.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.c.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.1.g.1 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
40.144.5.s.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.br.2 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
40.144.5.db.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.de.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.io.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.is.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.iw.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.jf.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.360.13.cc.1 | $40$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
120.144.5.chk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cho.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cim.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ciq.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egz.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eib.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eie.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.um.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.ieq.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bo.1 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bgc.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgd.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgq.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgr.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bos.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bot.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpg.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bph.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |