Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $160$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $15 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $8$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}\cdot10^{4}\cdot20^{2}\cdot40^{4}$ | Cusp orbits | $1^{8}\cdot2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40X15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.576.15.100 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&21\\0&37\end{bmatrix}$, $\begin{bmatrix}9&24\\0&33\end{bmatrix}$, $\begin{bmatrix}21&24\\0&17\end{bmatrix}$, $\begin{bmatrix}31&1\\0&13\end{bmatrix}$, $\begin{bmatrix}39&19\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | Group 1280.1114100 |
| Contains $-I$: | no $\quad$ (see 40.288.15.ef.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $1$ |
| Cyclic 40-torsion field degree: | $8$ |
| Full 40-torsion field degree: | $1280$ |
Jacobian
| Conductor: | $2^{52}\cdot5^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot4^{2}$ |
| Newforms: | 20.2.a.a$^{3}$, 40.2.a.a$^{2}$, 40.2.d.a, 80.2.a.a, 80.2.a.b, 160.2.d.a |
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 |
| 8.96.0-8.n.1.6 | $8$ | $6$ | $6$ | $0$ | $0$ | 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.n.1.6 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.288.7-40.cl.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $4^{2}$ |
| 40.288.7-40.cl.1.16 | $40$ | $2$ | $2$ | $7$ | $0$ | $4^{2}$ |
| 40.288.7-40.fn.1.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.fn.1.11 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.fs.2.12 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.fs.2.22 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1152.29-40.lz.1.20 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.lz.2.20 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mb.3.12 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mb.4.12 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mn.3.2 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mn.4.3 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mp.1.1 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.mp.2.1 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.33-40.bjv.2.1 | $40$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.bjy.1.5 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.bka.2.5 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.bkc.1.7 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.bke.2.7 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.bke.4.6 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.bkg.1.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.bkg.3.8 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.2880.91-40.ta.2.4 | $40$ | $5$ | $5$ | $91$ | $7$ | $1^{36}\cdot2^{8}\cdot4^{6}$ |