Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $800$ | ||
| 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 | $1^{4}\cdot2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-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: | 40.480.16.80 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&0\\36&17\end{bmatrix}$, $\begin{bmatrix}1&8\\4&1\end{bmatrix}$, $\begin{bmatrix}7&20\\0&7\end{bmatrix}$, $\begin{bmatrix}25&32\\26&11\end{bmatrix}$, $\begin{bmatrix}33&0\\34&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.16.d.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{50}\cdot5^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}\cdot2^{4}$ |
| Newforms: | 50.2.a.b$^{4}$, 200.2.a.e$^{2}$, 200.2.d.a, 200.2.d.c, 400.2.a.a, 400.2.a.f, 800.2.d.a, 800.2.d.c |
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.b.1.2 | $8$ | $5$ | $5$ | $0$ | $0$ | 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.b.1.2 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.240.8-20.b.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $2^{4}$ |
| 40.240.8-20.b.1.12 | $40$ | $2$ | $2$ | $8$ | $1$ | $2^{4}$ |
| 40.240.8-40.k.1.7 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.k.1.17 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.n.2.12 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
| 40.240.8-40.n.2.23 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{4}\cdot2^{2}$ |
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.bs.2.5 | $40$ | $2$ | $2$ | $33$ | $11$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.bv.1.2 | $40$ | $2$ | $2$ | $33$ | $9$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cb.2.3 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cc.1.3 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.ci.1.9 | $40$ | $2$ | $2$ | $33$ | $9$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cj.1.9 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.ck.1.9 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cl.1.5 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.co.1.1 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cp.2.3 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.ct.1.1 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{7}\cdot2^{5}$ |
| 40.960.33-40.cw.2.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{7}\cdot2^{5}$ |
| 40.960.35-40.w.2.5 | $40$ | $2$ | $2$ | $35$ | $7$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.ba.2.5 | $40$ | $2$ | $2$ | $35$ | $5$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.bg.2.5 | $40$ | $2$ | $2$ | $35$ | $7$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.960.35-40.bh.2.5 | $40$ | $2$ | $2$ | $35$ | $6$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.1440.46-40.h.2.21 | $40$ | $3$ | $3$ | $46$ | $4$ | $1^{14}\cdot4^{4}$ |
| 40.1920.61-40.bd.2.11 | $40$ | $4$ | $4$ | $61$ | $9$ | $1^{21}\cdot2^{4}\cdot4^{4}$ |