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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.93 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&8\\7&27\end{bmatrix}$, $\begin{bmatrix}17&6\\35&11\end{bmatrix}$, $\begin{bmatrix}17&22\\35&23\end{bmatrix}$, $\begin{bmatrix}37&18\\10&7\end{bmatrix}$ |
| 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 x y - z w $ |
| $=$ | $80 x^{2} - 10 y^{2} + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 10 x^{4} - 25 x^{2} y^{2} + x^{2} z^{2} - 5 y^{2} z^{2} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{5600y^{2}z^{4}+10800y^{2}z^{2}w^{2}+4850y^{2}w^{4}-1728z^{6}-4496z^{4}w^{2}-3124z^{2}w^{4}-27w^{6}}{160y^{2}z^{4}-80y^{2}z^{2}w^{2}-10y^{2}w^{4}-64z^{6}+16z^{4}w^{2}-12z^{2}w^{4}-w^{6}}$ |
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 |
| 40.12.0.bt.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bv.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.i.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.cu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.eb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ek.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fj.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.he.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.ct.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.ex.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.pl.1 | $40$ | $10$ | $10$ | $17$ | $7$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.pz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.qd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.wj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ya.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.yc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.hj.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.ed.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.sf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.sz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.vx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.wb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.ed.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |