Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $80$ | ||
Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
Genus: | $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{4}$ | Cusp orbits | $4$ | ||
Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.40.1.53 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&9\\37&28\end{bmatrix}$, $\begin{bmatrix}13&23\\20&27\end{bmatrix}$, $\begin{bmatrix}33&2\\39&7\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $72$ |
Cyclic 40-torsion field degree: | $1152$ |
Full 40-torsion field degree: | $18432$ |
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 $ | $=$ | $ 10 x^{2} + 4 y^{2} + 3 z^{2} + z w + w^{2} $ |
$=$ | $10 x^{2} - 6 y^{2} - 4 z^{2} - z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 70 x^{2} y^{2} + 10 x^{2} z^{2} + 1225 y^{4} - 300 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -5^3\,\frac{(z+2w)(z+3w)^{3}(3z^{2}+3zw+2w^{2})^{3}}{(z^{2}+zw-w^{2})^{5}}$ |
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 |
---|---|---|---|---|---|---|
20.20.1.b.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.20.0.d.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.20.0.h.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 |
---|---|---|---|---|---|---|
40.120.5.en.1 | $40$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
40.120.5.fm.1 | $40$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
40.160.9.bt.1 | $40$ | $4$ | $4$ | $9$ | $4$ | $1^{8}$ |
120.120.9.bof.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.160.9.jd.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
200.200.9.bp.1 | $200$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.320.21.dp.1 | $280$ | $8$ | $8$ | $21$ | $?$ | not computed |