Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.382 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&32\\0&37\end{bmatrix}$, $\begin{bmatrix}13&28\\11&15\end{bmatrix}$, $\begin{bmatrix}33&14\\11&15\end{bmatrix}$ |
| 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: | $15360$ |
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} - z^{2} - w^{2} $ |
| $=$ | $10 y^{2} - 3 z^{2} + 2 z w - 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 12 x^{2} y^{2} + 36 y^{4} - 20 y^{2} z^{2} + 25 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(3z^{2}+2zw+3w^{2})^{3}(5z^{2}-2zw+5w^{2})^{3}}{(z-w)^{8}(z^{2}+w^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.1.bd.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.0.cy.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dc.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.eb.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.eg.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.bf.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bu.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 |
|---|---|---|---|---|---|---|
| 40.240.17.lj.1 | $40$ | $5$ | $5$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.288.17.bdj.1 | $40$ | $6$ | $6$ | $17$ | $7$ | $1^{14}\cdot2$ |
| 40.480.33.btr.1 | $40$ | $10$ | $10$ | $33$ | $9$ | $1^{28}\cdot2^{2}$ |
| 80.96.1.dl.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.96.1.dl.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.9.fiz.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.buj.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.1.li.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1.li.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |