Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $16 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $20^{8}\cdot40^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $5$ | ||||||
| $\overline{\Q}$-gonality: | $5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40A16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.480.16.2 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&4\\8&13\end{bmatrix}$, $\begin{bmatrix}1&8\\36&27\end{bmatrix}$, $\begin{bmatrix}17&16\\4&13\end{bmatrix}$, $\begin{bmatrix}33&24\\4&23\end{bmatrix}$, $\begin{bmatrix}33&24\\36&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.16.f.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{42}\cdot5^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}\cdot2^{4}$ |
| Newforms: | 50.2.a.b$^{4}$, 200.2.a.e$^{2}$, 200.2.d.a$^{2}$, 200.2.d.c$^{2}$, 400.2.a.a, 400.2.a.f |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.c.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.c.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 20.240.8-20.b.1.1 | $20$ | $2$ | $2$ | $8$ | $1$ | $2^{4}$ |
| 40.240.8-20.b.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $2^{4}$ |
| 40.240.8-40.n.1.1 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.n.1.29 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.n.2.5 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.n.2.31 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
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.ct.1.2 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.ct.2.5 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cu.1.3 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cu.2.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cv.1.1 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cv.2.5 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cw.1.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cw.2.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{7}\cdot2^{5}$ |
| 40.960.35-40.bm.1.1 | $40$ | $2$ | $2$ | $35$ | $5$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.bn.1.1 | $40$ | $2$ | $2$ | $35$ | $3$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.bo.1.1 | $40$ | $2$ | $2$ | $35$ | $5$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.bp.1.1 | $40$ | $2$ | $2$ | $35$ | $4$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.1440.46-40.l.1.2 | $40$ | $3$ | $3$ | $46$ | $2$ | $1^{14}\cdot4^{4}$ |
| 40.1920.61-40.bf.1.4 | $40$ | $4$ | $4$ | $61$ | $5$ | $1^{21}\cdot2^{4}\cdot4^{4}$ |