Invariants
| Level: | $22$ | $\SL_2$-level: | $22$ | Newform level: | $484$ | ||
| Index: | $396$ | $\PSL_2$-index: | $396$ | ||||
| Genus: | $22 = 1 + \frac{ 396 }{12} - \frac{ 12 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (of which $3$ are rational) | Cusp widths | $22^{18}$ | Cusp orbits | $1^{3}\cdot5\cdot10$ | ||
| Elliptic points: | $12$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 12$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7$) |
Other labels
| Cummins and Pauli (CP) label: | 22B22 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 22.396.22.3 |
Level structure
| $\GL_2(\Z/22\Z)$-generators: | $\begin{bmatrix}0&13\\1&0\end{bmatrix}$, $\begin{bmatrix}0&13\\17&0\end{bmatrix}$ |
| $\GL_2(\Z/22\Z)$-subgroup: | $C_{10}\wr C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 22-isogeny field degree: | $2$ |
| Cyclic 22-torsion field degree: | $20$ |
| Full 22-torsion field degree: | $200$ |
Jacobian
| Conductor: | $2^{18}\cdot11^{40}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{12}\cdot2^{5}$ |
| Newforms: | 11.2.a.a$^{3}$, 44.2.a.a, 121.2.a.a, 121.2.a.b$^{2}$, 121.2.a.c, 121.2.a.d, 242.2.a.a, 242.2.a.b, 242.2.a.c, 242.2.a.d, 242.2.a.e, 242.2.a.f, 484.2.a.a, 484.2.a.c |
Models
Some stored models are too large to be displayed and are available for download.
Rational points
This modular curve has 3 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Canonical model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:0:-1:0:0:0:0:0:0:1:2:5:0:-2:2:0:2:1:-2:2:-1:1)$, $(0:2/3:0:1/3:-2/3:1:-1/3:1:0:1/3:1/3:-1/3:-1/3:-1/3:1:0:2/3:-1/3:-2/3:-2/3:-2/3:1)$, $(0:-2/3:1/3:-1/3:2/3:-1:1/3:-1:0:1/3:-4/3:0:0:-1/3:0:0:2/3:-1/3:1/3:1:0:0)$ | ||
| 49.a2 | $-7$ | $-3375$ | $= -1 \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(2/7:0:-1/7:0:-4/7:-1/7:0:-1/7:3/7:-2/7:2/7:1/7:-1/7:-1/7:0:-1/7:1/7:-5/7:0:-1/7:2/7:1)$, $(-2:0:-3:0:4:1:0:1:-3:-2:5:1:0:0:2:1:2:-3:0:1:2:1)$ |
Maps to other modular curves
Some stored maps are too large to be displayed and are available for download.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 22.132.6.b.1 | $22$ | $3$ | $3$ | $6$ | $4$ | $1^{8}\cdot2^{4}$ |
| 22.198.10.a.1 | $22$ | $2$ | $2$ | $10$ | $4$ | $1^{8}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(22)$ | $22$ | $2$ | $2$ | $49$ | $12$ | $1^{13}\cdot2^{7}$ |
| 22.792.49.d.1 | $22$ | $2$ | $2$ | $49$ | $15$ | $1^{13}\cdot2^{7}$ |
| 22.1980.118.a.1 | $22$ | $5$ | $5$ | $118$ | $32$ | $1^{42}\cdot2^{27}$ |
| 44.792.49.f.1 | $44$ | $2$ | $2$ | $49$ | $19$ | $1^{13}\cdot2^{7}$ |
| 44.792.49.x.1 | $44$ | $2$ | $2$ | $49$ | $22$ | $1^{13}\cdot2^{7}$ |
| 44.792.49.bh.1 | $44$ | $2$ | $2$ | $49$ | $16$ | $1^{11}\cdot2^{6}\cdot4$ |
| 44.792.49.bj.1 | $44$ | $2$ | $2$ | $49$ | $21$ | $1^{11}\cdot2^{6}\cdot4$ |
| 44.792.49.bp.1 | $44$ | $2$ | $2$ | $49$ | $18$ | $1^{11}\cdot2^{6}\cdot4$ |
| 44.792.49.br.1 | $44$ | $2$ | $2$ | $49$ | $31$ | $1^{11}\cdot2^{6}\cdot4$ |
| 44.792.55.z.1 | $44$ | $2$ | $2$ | $55$ | $9$ | $1^{7}\cdot2^{11}\cdot4$ |
| 44.792.55.bb.1 | $44$ | $2$ | $2$ | $55$ | $18$ | $1^{7}\cdot2^{11}\cdot4$ |
| 44.792.55.bp.1 | $44$ | $2$ | $2$ | $55$ | $20$ | $1^{7}\cdot2^{11}\cdot4$ |
| 44.792.55.br.1 | $44$ | $2$ | $2$ | $55$ | $21$ | $1^{7}\cdot2^{11}\cdot4$ |
| 66.792.49.f.1 | $66$ | $2$ | $2$ | $49$ | $23$ | $1^{13}\cdot2^{7}$ |
| 66.792.49.l.1 | $66$ | $2$ | $2$ | $49$ | $22$ | $1^{13}\cdot2^{7}$ |
| 66.1188.82.h.1 | $66$ | $3$ | $3$ | $82$ | $39$ | $1^{20}\cdot2^{12}\cdot4^{4}$ |
| 66.1584.115.h.1 | $66$ | $4$ | $4$ | $115$ | $35$ | $1^{43}\cdot2^{21}\cdot4^{2}$ |