Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| 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: | 40.24.1.68 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&22\\39&17\end{bmatrix}$, $\begin{bmatrix}21&22\\38&19\end{bmatrix}$, $\begin{bmatrix}23&32\\25&37\end{bmatrix}$, $\begin{bmatrix}31&18\\21&25\end{bmatrix}$ |
| 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: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} - z w $ |
| $=$ | $5 y^{2} - 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 5 y^{2} z^{2} - 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.12.0.l.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.1.g.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 |
|---|---|---|---|---|---|---|
| 40.48.1.ig.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ih.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ii.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ij.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ju.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jw.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jx.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.fe.1 | $40$ | $5$ | $5$ | $9$ | $6$ | $1^{6}\cdot2$ |
| 40.144.9.kk.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.240.17.bam.1 | $40$ | $10$ | $10$ | $17$ | $9$ | $1^{12}\cdot2^{2}$ |
| 80.48.3.bv.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bv.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.et.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.et.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.ev.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.ev.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.gz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.gz.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.1.clm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cln.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.clo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.clp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cms.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.che.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.vu.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 240.48.3.gj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.gj.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.kp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.kp.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.kr.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.kr.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.sj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.sj.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.48.1.btm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.btn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bto.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.btp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.buc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bud.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bue.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.buf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.mc.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |