Invariants
| Level: | $55$ | $\SL_2$-level: | $55$ | Newform level: | $3025$ | ||
| Index: | $6600$ | $\PSL_2$-index: | $3300$ | ||||
| Genus: | $246 = 1 + \frac{ 3300 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 60 }{2}$ | ||||||
| Cusps: | $60$ (none of which are rational) | Cusp widths | $55^{60}$ | Cusp orbits | $5^{2}\cdot10\cdot20^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $51$ | ||||||
| $\Q$-gonality: | $33 \le \gamma \le 55$ | ||||||
| $\overline{\Q}$-gonality: | $33 \le \gamma \le 55$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 55.6600.246.2 |
Level structure
| $\GL_2(\Z/55\Z)$-generators: | $\begin{bmatrix}28&3\\50&26\end{bmatrix}$, $\begin{bmatrix}38&28\\15&1\end{bmatrix}$ |
| $\GL_2(\Z/55\Z)$-subgroup: | $C_{20}\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 55.3300.246.a.1 for the level structure with $-I$) |
| Cyclic 55-isogeny field degree: | $12$ |
| Cyclic 55-torsion field degree: | $120$ |
| Full 55-torsion field degree: | $960$ |
Jacobian
| Conductor: | $5^{398}\cdot11^{492}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{5}\cdot3^{5}\cdot4^{10}\cdot6^{7}\cdot8^{7}\cdot12^{5}\cdot16$ |
| Newforms: | 121.2.a.a$^{3}$, 605.2.a.a$^{2}$, 605.2.a.e$^{2}$, 605.2.a.h$^{2}$, 605.2.a.j$^{2}$, 605.2.a.l$^{2}$, 605.2.a.m$^{2}$, 605.2.b.e$^{2}$, 605.2.b.g$^{2}$, 605.2.b.h$^{2}$, 3025.2.a.bd, 3025.2.a.bf, 3025.2.a.bg, 3025.2.a.bh, 3025.2.a.bi, 3025.2.a.bl, 3025.2.a.bn, 3025.2.a.bo, 3025.2.a.e, 3025.2.a.g, 3025.2.a.l, 3025.2.a.p, 3025.2.a.q, 3025.2.a.s, 3025.2.a.v, 3025.2.a.z, 3025.2.b.c, 3025.2.b.f, 3025.2.b.j, 3025.2.b.n, 3025.2.b.o, 3025.2.b.r, 3025.2.b.t, 3025.2.b.v, 3025.2.b.w, 3025.2.b.y |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7,13,\ldots,1481$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(11)$ | $11$ | $120$ | $60$ | $1$ | $0$ | $1^{6}\cdot2^{5}\cdot3^{5}\cdot4^{10}\cdot6^{7}\cdot8^{7}\cdot12^{5}\cdot16$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $55$ | $55$ | $0$ | $0$ | full Jacobian |
| 55.1320.46-55.a.1.3 | $55$ | $5$ | $5$ | $46$ | $7$ | $1^{4}\cdot2^{4}\cdot3^{4}\cdot4^{7}\cdot6^{6}\cdot8^{6}\cdot12^{4}\cdot16$ |
| 55.1320.46-55.a.2.2 | $55$ | $5$ | $5$ | $46$ | $7$ | $1^{4}\cdot2^{4}\cdot3^{4}\cdot4^{7}\cdot6^{6}\cdot8^{6}\cdot12^{4}\cdot16$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 55.19800.736-55.a.1.7 | $55$ | $3$ | $3$ | $736$ | $152$ | $1^{22}\cdot2^{35}\cdot4^{37}\cdot6^{5}\cdot8^{16}\cdot12^{5}\cdot16^{2}$ |
| 55.26400.981-55.a.1.6 | $55$ | $4$ | $4$ | $981$ | $198$ | $1^{46}\cdot2^{43}\cdot3^{15}\cdot4^{52}\cdot6^{11}\cdot8^{22}\cdot12^{5}\cdot16^{3}$ |