Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.97 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&30\\16&23\end{bmatrix}$, $\begin{bmatrix}21&10\\37&7\end{bmatrix}$, $\begin{bmatrix}37&14\\16&39\end{bmatrix}$, $\begin{bmatrix}39&10\\17&37\end{bmatrix}$ |
| 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: | $30720$ |
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 $ | $=$ | $ 10 y^{2} - 4 z^{2} + w^{2} $ |
| $=$ | $20 x^{2} + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 10 y^{2} z^{2} - 100 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{20}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z-w)^{3}(z+w)^{3}}{w^{2}z^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.12.0.bk.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.cb.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 |
|---|---|---|---|---|---|---|
| 40.48.1.fg.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fi.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fj.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.go.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gp.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gq.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gr.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.br.1 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.144.9.cp.1 | $40$ | $6$ | $6$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.240.17.lx.1 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.ry.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sa.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.te.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.th.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.cx.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.cb.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.oo.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.op.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.oq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.or.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pe.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pg.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ph.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.cb.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |