Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.631 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&24\\29&3\end{bmatrix}$, $\begin{bmatrix}9&14\\4&11\end{bmatrix}$, $\begin{bmatrix}21&14\\1&27\end{bmatrix}$ |
| 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: | $7680$ |
Jacobian
| Conductor: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y - y^{2} + z w $ |
| $=$ | $3 x^{2} + 2 x y + 2 y^{2} + z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{3} z + 15 x^{2} y^{2} + 7 x^{2} z^{2} - 20 x y^{2} z + 6 x z^{3} + 25 y^{4} - 15 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{(z^{4}-10z^{2}w^{2}+w^{4})^{3}(z^{4}+6z^{2}w^{2}+w^{4})^{3}}{w^{4}z^{4}(z-w)^{8}(z+w)^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.48.0.q.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0.bs.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1.ib.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 |
|---|---|---|---|---|---|---|
| 40.480.33.dwb.2 | $40$ | $5$ | $5$ | $33$ | $7$ | $1^{14}\cdot2^{9}$ |
| 40.576.33.crv.2 | $40$ | $6$ | $6$ | $33$ | $4$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.960.65.eaz.2 | $40$ | $10$ | $10$ | $65$ | $9$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 120.288.17.ddcj.2 | $120$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 120.384.17.fkq.2 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |