Invariants
| Level: | $45$ | $\SL_2$-level: | $45$ | Newform level: | $675$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $24 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $15^{6}\cdot45^{6}$ | Cusp orbits | $2^{3}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 45A24 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 45.720.24.12 |
Level structure
| $\GL_2(\Z/45\Z)$-generators: | $\begin{bmatrix}6&17\\26&3\end{bmatrix}$, $\begin{bmatrix}43&2\\16&0\end{bmatrix}$, $\begin{bmatrix}43&6\\25&2\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 45.360.24.a.2 for the level structure with $-I$) |
| Cyclic 45-isogeny field degree: | $18$ |
| Cyclic 45-torsion field degree: | $216$ |
| Full 45-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $3^{52}\cdot5^{48}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}\cdot2^{3}\cdot6\cdot8$ |
| Newforms: | 75.2.a.a, 75.2.a.b, 225.2.e.a, 225.2.e.b, 225.2.e.e, 675.2.a.b, 675.2.a.h, 675.2.a.k, 675.2.a.p |
Rational points
This modular curve has no $\Q_p$ points for $p=2$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 45.72.0-9.f.1.1 | $45$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 45.240.8-45.a.1.5 | $45$ | $3$ | $3$ | $8$ | $3$ | $2\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 45.1440.47-45.a.2.1 | $45$ | $2$ | $2$ | $47$ | $6$ | $1^{3}\cdot2^{2}\cdot8^{2}$ |
| 45.1440.47-45.b.1.3 | $45$ | $2$ | $2$ | $47$ | $7$ | $1^{3}\cdot2^{2}\cdot8^{2}$ |
| 45.1440.47-45.g.1.3 | $45$ | $2$ | $2$ | $47$ | $6$ | $1^{3}\cdot2^{2}\cdot8^{2}$ |
| 45.1440.47-45.h.2.1 | $45$ | $2$ | $2$ | $47$ | $6$ | $1^{3}\cdot2^{2}\cdot8^{2}$ |
| 45.2160.73-45.a.2.3 | $45$ | $3$ | $3$ | $73$ | $6$ | $1^{15}\cdot2^{3}\cdot6^{2}\cdot8^{2}$ |
| 45.2160.73-45.s.2.5 | $45$ | $3$ | $3$ | $73$ | $8$ | $1^{5}\cdot2^{8}\cdot6^{2}\cdot8^{2}$ |