Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
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 | $2^{2}\cdot10^{2}$ | 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: | 10D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.155 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}27&27\\9&10\end{bmatrix}$, $\begin{bmatrix}30&29\\1&8\end{bmatrix}$, $\begin{bmatrix}37&16\\6&27\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $30720$ |
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 $ | $=$ | $ 2 x^{2} + y z $ |
$=$ | $8 x^{2} + 25 y^{2} - 18 y z + 5 z^{2} + 10 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} + 44 x^{2} z^{2} + 10 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 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{14770296yz^{5}-391419000yz^{3}w^{2}+5343750yzw^{4}-5559840z^{6}+268262820z^{4}w^{2}-51547500z^{2}w^{4}+78125w^{6}}{z(68381yz^{4}-300500yz^{2}w^{2}+62500yw^{4}-25740z^{5}+3520z^{3}w^{2}+110000zw^{4})}$ |
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 |
---|---|---|---|---|---|---|
20.12.1.b.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.12.0.bo.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.0.bw.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 |
---|---|---|---|---|---|---|
40.72.1.be.1 | $40$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
40.96.5.s.2 | $40$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.120.5.cv.1 | $40$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
120.72.5.bbe.2 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.ni.2 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
200.120.5.s.1 | $200$ | $5$ | $5$ | $5$ | $?$ | not computed |
280.192.13.fu.2 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |