Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.856 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&34\\12&21\end{bmatrix}$, $\begin{bmatrix}11&30\\12&29\end{bmatrix}$, $\begin{bmatrix}39&18\\36&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.z.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3840$ |
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} + 2 z^{2} - 2 w^{2} $ |
| $=$ | $10 x^{2} + 2 z^{2} - w^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(16z^{8}-32z^{6}w^{2}+20z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z-w)^{2}(z+w)^{2}(2z^{2}-w^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.k.2.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.0-40.o.2.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.o.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.p.2.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.p.2.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.s.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.t.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.t.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-8.k.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.q.1.13 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.q.1.16 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.r.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.r.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.cy.2.5 | $40$ | $5$ | $5$ | $33$ | $5$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.kc.2.13 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.ny.2.3 | $40$ | $10$ | $10$ | $65$ | $11$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 80.384.5-80.i.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.k.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bg.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bm.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.en.1.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.et.1.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.fp.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.fr.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.9-80.jt.2.4 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.ju.1.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.ka.2.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.kb.1.3 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-240.bg.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bi.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ds.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dy.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.od.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oj.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qt.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qv.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-240.bjr.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bjs.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bkg.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bkh.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |