Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $160$ | ||
| 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 $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot20^{4}\cdot40^{4}$ | Cusp orbits | $1^{4}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40AE17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.576.17.125 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&32\\38&15\end{bmatrix}$, $\begin{bmatrix}9&0\\24&17\end{bmatrix}$, $\begin{bmatrix}21&4\\26&21\end{bmatrix}$, $\begin{bmatrix}27&36\\38&15\end{bmatrix}$, $\begin{bmatrix}33&20\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_{10}:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 40.288.17.cc.2 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: | $2^{69}\cdot5^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2\cdot4^{2}$ |
| Newforms: | 20.2.a.a$^{2}$, 32.2.a.a$^{2}$, 40.2.a.a, 40.2.d.a, 160.2.a.a, 160.2.a.b, 160.2.a.c, 160.2.d.a |
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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.i.2.5 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.288.7-40.q.1.14 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.q.1.41 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.v.2.11 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.7-40.v.2.34 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.288.9-40.c.1.3 | $40$ | $2$ | $2$ | $9$ | $1$ | $4^{2}$ |
| 40.288.9-40.c.1.47 | $40$ | $2$ | $2$ | $9$ | $1$ | $4^{2}$ |
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.ec.1.13 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.ec.3.10 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.er.2.6 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.er.3.6 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.gz.1.11 | $40$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.he.2.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.hl.2.5 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.ho.1.11 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.ia.3.11 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.ia.4.6 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.ic.1.11 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.ic.2.6 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.jd.1.14 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.jd.2.12 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.jf.3.11 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.jf.4.6 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.1152.33-40.jo.1.11 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.jr.2.9 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.jy.2.5 | $40$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.kd.1.13 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot4^{2}$ |
| 40.1152.33-40.ni.1.1 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.ni.3.1 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.nt.2.1 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.1152.33-40.nt.4.1 | $40$ | $2$ | $2$ | $33$ | $1$ | $2^{2}\cdot4^{3}$ |
| 40.2880.97-40.gn.2.5 | $40$ | $5$ | $5$ | $97$ | $12$ | $1^{28}\cdot2^{14}\cdot4^{6}$ |