Invariants
| Level: | $9$ | $\SL_2$-level: | $9$ | Newform level: | $27$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $3\cdot9$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 9A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 9.12.1.1 |
Level structure
Jacobian
| Conductor: | $3^{3}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 27.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} $ |
Rational points
This modular curve has 2 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 | Weierstrass model | |
|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(0:-1:1)$ | |
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(0:0:1)$ | ||
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{(y+z)^{2}(9y+z)^{3}}{z^{3}y(y+z)}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 9.36.1.a.1 | $9$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 9.36.1.b.1 | $9$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 9.36.1.b.2 | $9$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 9.36.1.c.1 | $9$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 18.24.2.a.1 | $18$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 18.24.2.b.1 | $18$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 18.36.2.c.1 | $18$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 36.24.2.a.1 | $36$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 36.24.2.b.1 | $36$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 36.48.4.c.1 | $36$ | $4$ | $4$ | $4$ | $0$ | $1^{3}$ |
| 45.60.5.a.1 | $45$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 45.72.5.a.1 | $45$ | $6$ | $6$ | $5$ | $1$ | $1^{4}$ |
| 45.120.9.a.1 | $45$ | $10$ | $10$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 63.36.1.a.1 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.36.1.a.2 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.36.1.b.1 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.36.1.b.2 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.36.1.c.1 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.36.1.c.2 | $63$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 63.96.7.a.1 | $63$ | $8$ | $8$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 63.252.19.a.1 | $63$ | $21$ | $21$ | $19$ | $7$ | $1^{4}\cdot2^{3}\cdot4^{2}$ |
| 63.336.25.a.1 | $63$ | $28$ | $28$ | $25$ | $7$ | $1^{6}\cdot2^{5}\cdot4^{2}$ |
| 72.24.2.a.1 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.24.2.b.1 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.24.2.c.1 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.24.2.d.1 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 90.24.2.a.1 | $90$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 90.24.2.b.1 | $90$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 99.144.11.a.1 | $99$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 117.36.1.a.1 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.36.1.a.2 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.36.1.b.1 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.36.1.b.2 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.36.1.c.1 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.36.1.c.2 | $117$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 117.168.13.a.1 | $117$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 126.24.2.a.1 | $126$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 126.24.2.b.1 | $126$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 153.216.17.a.1 | $153$ | $18$ | $18$ | $17$ | $?$ | not computed |
| 171.36.1.a.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.36.1.a.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.36.1.b.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.36.1.b.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.36.1.c.1 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.36.1.c.2 | $171$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 171.240.19.a.1 | $171$ | $20$ | $20$ | $19$ | $?$ | not computed |
| 180.24.2.a.1 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.24.2.b.1 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 198.24.2.a.1 | $198$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 198.24.2.b.1 | $198$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 207.288.23.a.1 | $207$ | $24$ | $24$ | $23$ | $?$ | not computed |
| 234.24.2.a.1 | $234$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 234.24.2.b.1 | $234$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.24.2.a.1 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.24.2.b.1 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 279.36.1.a.1 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 279.36.1.a.2 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 279.36.1.b.1 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 279.36.1.b.2 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 279.36.1.c.1 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 279.36.1.c.2 | $279$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 306.24.2.a.1 | $306$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 306.24.2.b.1 | $306$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 333.36.1.a.1 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 333.36.1.a.2 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 333.36.1.b.1 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 333.36.1.b.2 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 333.36.1.c.1 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 333.36.1.c.2 | $333$ | $3$ | $3$ | $1$ | $?$ | dimension zero |