Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | 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 | $2^{6}\cdot10^{6}$ | 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: | 10K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.102 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&11\\34&33\end{bmatrix}$, $\begin{bmatrix}17&1\\38&15\end{bmatrix}$, $\begin{bmatrix}23&25\\20&23\end{bmatrix}$, $\begin{bmatrix}35&28\\2&21\end{bmatrix}$ |
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 $ | $=$ | $ x^{2} + x w + y^{2} $ |
$=$ | $ - 4 x w + 4 y^{2} - 10 z^{2} - 5 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 10 x^{2} y^{2} + 2 x^{2} z^{2} + 5 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{76800000000xz^{16}w+51200000000xz^{14}w^{3}+14144000000xz^{12}w^{5}+2073600000xz^{10}w^{7}+169600000xz^{8}w^{9}+7072000xz^{6}w^{11}+72000xz^{4}w^{13}-4320xz^{2}w^{15}-108xw^{17}+64000000000z^{18}+57600000000z^{16}w^{2}+12800000000z^{14}w^{4}-1648000000z^{12}w^{6}-1200000000z^{10}w^{8}-227840000z^{8}w^{10}-22196000z^{6}w^{12}-1206000z^{4}w^{14}-34560z^{2}w^{16}-405w^{18}}{w^{10}(10z^{2}+w^{2})^{2}(80xz^{2}w+4xw^{3}+400z^{4}+230z^{2}w^{2}+15w^{4})}$ |
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.d.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.cq.2 | $40$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
40.36.0.a.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.1.f.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.ey.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.fc.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ga.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.ge.2 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
40.144.5.ij.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.im.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.jl.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.jo.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.360.13.ck.1 | $40$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
120.144.5.cmo.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cms.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cos.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cow.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.edn.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.edq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.efr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.efu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.tm.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.iec.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bg.2 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bdt.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bdu.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.beh.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bei.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmj.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmk.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmx.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bmy.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |