Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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 | $4^{2}$ | ||
| 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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.409 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&38\\39&5\end{bmatrix}$, $\begin{bmatrix}25&22\\12&3\end{bmatrix}$, $\begin{bmatrix}39&20\\29&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 80.96.1-40.gd.1.1, 80.96.1-40.gd.1.2, 80.96.1-40.gd.1.3, 80.96.1-40.gd.1.4, 240.96.1-40.gd.1.1, 240.96.1-40.gd.1.2, 240.96.1-40.gd.1.3, 240.96.1-40.gd.1.4 |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 y^{2} - 2 z^{2} - w^{2} $ |
| $=$ | $10 x^{2} + z^{2} + w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(z^{2}+2w^{2})^{3}(3z^{2}+2w^{2})^{3}}{z^{8}(z^{2}+w^{2})^{2}}$ |
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.z.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.0.cl.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.cq.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dt.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.eh.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.y.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bj.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.kh.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.288.17.bbb.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.480.33.brj.1 | $40$ | $10$ | $10$ | $33$ | $8$ | $1^{28}\cdot2^{2}$ |
| 120.144.9.fev.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.bsr.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |