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: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.40.1.56 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}4&7\\13&2\end{bmatrix}$, $\begin{bmatrix}15&19\\14&15\end{bmatrix}$, $\begin{bmatrix}37&35\\13&28\end{bmatrix}$ |
| 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} - 3 y^{2} - y z - z^{2} + 4 w^{2} $ |
| $=$ | $10 x^{2} + 4 y^{2} + y z + 2 z^{2} - 6 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 20 x^{4} - 100 x^{2} y^{2} - 10 x^{2} z^{2} + 225 y^{4} + 30 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\cdot5^2\,\frac{1801656yz^{9}+904568yz^{7}w^{2}-96332922yz^{5}w^{4}-108353700yz^{3}w^{6}-11409552yzw^{8}+116208z^{10}+13996884z^{8}w^{2}-30432136z^{6}w^{4}-169518195z^{4}w^{6}-59603796z^{2}w^{8}-1296540w^{10}}{208525yz^{9}-706300yz^{7}w^{2}+994700yz^{5}w^{4}-686000yz^{3}w^{6}+192080yzw^{8}+13450z^{10}-168150z^{8}w^{2}+313600z^{6}w^{4}-127400z^{4}w^{6}-109760z^{2}w^{8}+76832w^{10}}$ |
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.c.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.20.0.g.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.ek.1 | $40$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 40.120.5.fd.1 | $40$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 40.160.9.bq.1 | $40$ | $4$ | $4$ | $9$ | $4$ | $1^{8}$ |
| 120.120.9.boc.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.160.9.ja.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 200.200.9.bm.1 | $200$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.320.21.dm.1 | $280$ | $8$ | $8$ | $21$ | $?$ | not computed |