Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $13 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (of which $2$ are rational) | Cusp widths | $10^{36}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4^{3}\cdot8^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10A13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.720.13.18 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&2\\20&23\end{bmatrix}$, $\begin{bmatrix}11&26\\30&27\end{bmatrix}$, $\begin{bmatrix}19&9\\0&33\end{bmatrix}$, $\begin{bmatrix}29&5\\0&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.360.13.cl.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{58}\cdot5^{25}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{3}$ |
| Newforms: | 50.2.a.a, 50.2.a.b, 50.2.b.a, 320.2.a.a, 1600.2.a.i, 1600.2.a.p, 1600.2.a.w$^{2}$, 1600.2.c.e, 1600.2.c.h |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|---|
| 10.360.4-10.a.1.3 | $10$ | $2$ | $2$ | $4$ | $0$ | $1^{5}\cdot2^{2}$ |
| 40.144.1-40.ba.1.3 | $40$ | $5$ | $5$ | $1$ | $0$ | $1^{6}\cdot2^{3}$ |
| 40.144.1-40.ba.2.3 | $40$ | $5$ | $5$ | $1$ | $0$ | $1^{6}\cdot2^{3}$ |
| 40.240.5-40.cx.1.3 | $40$ | $3$ | $3$ | $5$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.360.4-10.a.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{5}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1440.37-40.kd.1.1 | $40$ | $2$ | $2$ | $37$ | $7$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.kj.1.2 | $40$ | $2$ | $2$ | $37$ | $7$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.lf.1.2 | $40$ | $2$ | $2$ | $37$ | $5$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.ll.1.2 | $40$ | $2$ | $2$ | $37$ | $6$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.pr.1.2 | $40$ | $2$ | $2$ | $37$ | $5$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.pz.1.1 | $40$ | $2$ | $2$ | $37$ | $5$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.qt.1.1 | $40$ | $2$ | $2$ | $37$ | $7$ | $1^{12}\cdot2^{6}$ |
| 40.1440.37-40.rb.1.1 | $40$ | $2$ | $2$ | $37$ | $7$ | $1^{12}\cdot2^{6}$ |