Invariants
| Level: | $55$ | $\SL_2$-level: | $55$ | Newform level: | $605$ | ||
| Index: | $1320$ | $\PSL_2$-index: | $660$ | ||||
| Genus: | $46 = 1 + \frac{ 660 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $11^{10}\cdot55^{10}$ | Cusp orbits | $5^{2}\cdot10$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 24$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 55.1320.46.2 |
Level structure
| $\GL_2(\Z/55\Z)$-generators: | $\begin{bmatrix}8&23\\51&40\end{bmatrix}$, $\begin{bmatrix}49&7\\15&6\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 55.660.46.a.2 for the level structure with $-I$) |
| Cyclic 55-isogeny field degree: | $12$ |
| Cyclic 55-torsion field degree: | $120$ |
| Full 55-torsion field degree: | $4800$ |
Jacobian
| Conductor: | $5^{44}\cdot11^{92}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2\cdot3\cdot4^{3}\cdot6\cdot8\cdot12$ |
| Newforms: | 121.2.a.a$^{2}$, 605.2.a.a, 605.2.a.e, 605.2.a.h, 605.2.a.j, 605.2.a.l, 605.2.a.m, 605.2.b.e, 605.2.b.g, 605.2.b.h |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7,13,17,47$, and therefore no 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 |
|---|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(11)$ | $11$ | $24$ | $12$ | $1$ | $0$ | $1^{2}\cdot2\cdot3\cdot4^{3}\cdot6\cdot8\cdot12$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 55.3960.136-55.a.2.4 | $55$ | $3$ | $3$ | $136$ | $22$ | $1^{10}\cdot2^{7}\cdot4^{8}\cdot6\cdot8^{2}\cdot12$ |
| 55.5280.181-55.a.2.2 | $55$ | $4$ | $4$ | $181$ | $27$ | $1^{20}\cdot2^{6}\cdot3^{3}\cdot4^{13}\cdot6\cdot8^{3}\cdot12$ |
| 55.6600.246-55.a.1.4 | $55$ | $5$ | $5$ | $246$ | $51$ | $1^{4}\cdot2^{4}\cdot3^{4}\cdot4^{7}\cdot6^{6}\cdot8^{6}\cdot12^{4}\cdot16$ |