Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| 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 | $4^{8}\cdot8^{2}\cdot20^{8}\cdot40^{2}$ | Cusp orbits | $1^{8}\cdot2^{6}$ | ||
| 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: | 40V15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.576.15.957 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&0\\12&3\end{bmatrix}$, $\begin{bmatrix}15&8\\28&23\end{bmatrix}$, $\begin{bmatrix}17&4\\28&33\end{bmatrix}$, $\begin{bmatrix}17&20\\12&23\end{bmatrix}$, $\begin{bmatrix}31&12\\20&27\end{bmatrix}$, $\begin{bmatrix}33&36\\12&37\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_{10}.C_2^6$ |
| Contains $-I$: | no $\quad$ (see 40.288.15.i.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: | $2^{44}\cdot5^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot4^{2}$ |
| Newforms: | 20.2.a.a$^{3}$, 40.2.a.a$^{2}$, 40.2.d.a$^{2}$, 80.2.a.a, 80.2.a.b |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.96.0-8.c.1.4 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.288.7-20.b.1.14 | $40$ | $2$ | $2$ | $7$ | $0$ | $4^{2}$ |
| 40.288.7-20.b.1.21 | $40$ | $2$ | $2$ | $7$ | $0$ | $4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1152.29-40.t.1.30 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.t.2.32 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.x.1.18 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.x.2.19 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.bz.1.20 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.bz.2.20 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.cd.1.20 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.cd.2.18 | $40$ | $2$ | $2$ | $29$ | $0$ | $2^{3}\cdot4^{2}$ |
| 40.1152.33-40.jt.1.21 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jt.2.24 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.ju.1.21 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.ju.2.24 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jv.1.22 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jv.2.20 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jv.3.22 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jv.4.24 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jw.1.21 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jw.2.22 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jw.3.24 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jw.4.22 | $40$ | $2$ | $2$ | $33$ | $0$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.jx.1.19 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jx.2.23 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jy.1.22 | $40$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jy.2.23 | $40$ | $2$ | $2$ | $33$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.37-40.eq.1.5 | $40$ | $2$ | $2$ | $37$ | $2$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.eq.1.9 | $40$ | $2$ | $2$ | $37$ | $2$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.er.1.7 | $40$ | $2$ | $2$ | $37$ | $1$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.er.1.11 | $40$ | $2$ | $2$ | $37$ | $1$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.es.1.1 | $40$ | $2$ | $2$ | $37$ | $4$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.es.1.13 | $40$ | $2$ | $2$ | $37$ | $4$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.et.1.3 | $40$ | $2$ | $2$ | $37$ | $4$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.et.1.15 | $40$ | $2$ | $2$ | $37$ | $4$ | $1^{8}\cdot2^{3}\cdot4^{2}$ |
| 40.1152.37-40.eu.1.2 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.eu.1.14 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.eu.2.6 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.eu.2.10 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.ev.1.4 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.ev.1.16 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.ev.2.8 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.1152.37-40.ev.2.12 | $40$ | $2$ | $2$ | $37$ | $0$ | $2^{3}\cdot4^{2}\cdot8$ |
| 40.2880.91-40.k.1.5 | $40$ | $5$ | $5$ | $91$ | $5$ | $1^{36}\cdot2^{8}\cdot4^{6}$ |