Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $320$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $8$ are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot20^{4}\cdot40^{4}$ | Cusp orbits | $1^{8}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40AE17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.576.17.142 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&28\\8&39\end{bmatrix}$, $\begin{bmatrix}33&24\\18&7\end{bmatrix}$, $\begin{bmatrix}39&12\\2&29\end{bmatrix}$, $\begin{bmatrix}39&28\\14&5\end{bmatrix}$, $\begin{bmatrix}39&36\\28&11\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_{10}:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 40.288.17.cx.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $16$ |
| Full 40-torsion field degree: | $1280$ |
Jacobian
| Conductor: | $2^{67}\cdot5^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2\cdot4^{2}$ |
| Newforms: | 20.2.a.a$^{2}$, 40.2.a.a, 40.2.d.a$^{2}$, 64.2.a.a$^{2}$, 320.2.a.b, 320.2.a.e, 320.2.a.g |
Rational points
This modular curve has 8 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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.n.1.3 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.n.1.3 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.288.7-40.v.1.3 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.v.1.37 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.v.2.14 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.v.2.37 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.9-40.d.1.22 | $40$ | $2$ | $2$ | $9$ | $0$ | $4^{2}$ |
| 40.288.9-40.d.1.36 | $40$ | $2$ | $2$ | $9$ | $0$ | $4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1152.33-40.fr.1.18 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.fr.2.19 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.gk.1.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.gk.2.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.jt.1.3 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.jt.2.7 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.ko.1.3 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.ko.2.6 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.le.1.1 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.le.2.1 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.le.3.1 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.le.4.1 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.lu.1.4 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.lu.2.7 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.lu.3.5 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.lu.4.9 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.mg.1.3 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.mg.2.7 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.mn.1.3 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.mn.2.7 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.oj.1.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.oj.2.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.ou.1.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.ou.2.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{2}\cdot4^{3}$ |
| 40.2880.97-40.hw.1.8 | $40$ | $5$ | $5$ | $97$ | $9$ | $1^{28}\cdot2^{14}\cdot4^{6}$ |