Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{8}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.1156 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}23&16\\28&15\end{bmatrix}$, $\begin{bmatrix}31&16\\38&27\end{bmatrix}$, $\begin{bmatrix}33&36\\36&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.j.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $3840$ |
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 $ | $=$ | $ 3 x^{2} - 2 x y - w^{2} $ |
| $=$ | $x^{2} + 2 x y - 2 y^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{2} y^{2} - 6 x^{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 to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(z^{8}-8z^{6}w^{2}+20z^{4}w^{4}-16z^{2}w^{6}+16w^{8})^{3}}{w^{8}z^{4}(z-2w)^{2}(z+2w)^{2}(z^{2}-2w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.1.j.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{2}Y^{2}-6X^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.96.0-8.e.2.5 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.e.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.f.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.f.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.k.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.k.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.l.2.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.l.2.5 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-8.m.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.m.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.n.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.n.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.p.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.p.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.960.33-40.dt.1.4 | $40$ | $5$ | $5$ | $33$ | $7$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.mn.1.11 | $40$ | $6$ | $6$ | $33$ | $2$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.rz.2.6 | $40$ | $10$ | $10$ | $65$ | $12$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 80.384.5-16.e.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.n.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.q.2.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.r.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.t.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-16.y.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.be.2.7 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.dm.2.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ds.2.7 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.dt.2.7 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ea.2.12 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ey.2.11 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.m.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ca.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.cc.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.cd.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ck.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.da.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.di.2.18 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ki.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ko.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.kp.2.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lm.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ow.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |