Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.1282 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&32\\24&31\end{bmatrix}$, $\begin{bmatrix}25&4\\27&23\end{bmatrix}$, $\begin{bmatrix}27&14\\32&1\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} - 5 y^{2} + z^{2} - z w - w^{2} $ |
| $=$ | $4 x^{2} - 12 x y + 4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 8 x^{3} z + 200 x^{2} y^{2} + 34 x^{2} z^{2} - 100 x y^{2} z + 72 x z^{3} - 125 y^{4} + 81 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(3z^{4}+4z^{3}w+12z^{2}w^{2}-4zw^{3}+3w^{4})^{3}(7z^{4}-4z^{3}w+8z^{2}w^{2}+4zw^{3}+7w^{4})^{3}}{(z^{2}+w^{2})^{4}(z^{2}-zw-w^{2})^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 40.48.1.gl.1 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.bor.1 | $40$ | $5$ | $5$ | $33$ | $4$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
| 40.576.33.cjb.2 | $40$ | $6$ | $6$ | $33$ | $1$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.960.65.dfn.1 | $40$ | $10$ | $10$ | $65$ | $8$ | $1^{28}\cdot2^{6}\cdot4^{6}$ |
| 120.288.17.bfbn.1 | $120$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 120.384.17.fkb.2 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |