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: | 40.24.1.92 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&2\\10&21\end{bmatrix}$, $\begin{bmatrix}17&4\\15&3\end{bmatrix}$, $\begin{bmatrix}21&34\\13&39\end{bmatrix}$, $\begin{bmatrix}33&4\\30&17\end{bmatrix}$ |
| 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 $ | $=$ | $ 5 y^{2} - 4 z^{2} + w^{2} $ |
| $=$ | $10 x^{2} + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 5 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}w$ |
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.
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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 |
| 20.12.0.l.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.ca.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.fs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ft.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ha.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.bu.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.cs.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.ma.1 | $40$ | $10$ | $10$ | $17$ | $6$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.sk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ts.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.tt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.da.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.ce.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.pq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ps.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.pt.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qg.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qi.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.qj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.ce.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |