Invariants
| Level: | $36$ | $\SL_2$-level: | $18$ | Newform level: | $27$ | ||
| Index: | $24$ | $\PSL_2$-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: | 36.24.1.5 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}14&5\\15&11\end{bmatrix}$, $\begin{bmatrix}22&19\\33&32\end{bmatrix}$, $\begin{bmatrix}22&27\\9&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.12.1.a.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $18$ |
| Cyclic 36-torsion field degree: | $216$ |
| Full 36-torsion field degree: | $15552$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.8.0-3.a.1.4 | $12$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 36.48.2-18.a.1.1 | $36$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 36.48.2-36.a.1.5 | $36$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 36.48.2-18.b.1.2 | $36$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 36.48.2-36.b.1.4 | $36$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 36.72.1-9.a.1.3 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.72.1-9.b.1.4 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.72.1-9.b.2.3 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.72.1-9.c.1.3 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.72.2-18.c.1.12 | $36$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 36.96.4-36.c.1.6 | $36$ | $4$ | $4$ | $4$ | $0$ | $1^{3}$ |
| 72.48.2-72.a.1.6 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.b.1.5 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.c.1.7 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.d.1.7 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.48.2-90.a.1.4 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.48.2-180.a.1.5 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.48.2-90.b.1.3 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.48.2-180.b.1.7 | $180$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 180.120.5-45.a.1.5 | $180$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 180.144.5-45.a.1.15 | $180$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 180.240.9-45.a.1.14 | $180$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 252.48.2-126.a.1.2 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.48.2-252.a.1.5 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.48.2-126.b.1.4 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.48.2-252.b.1.6 | $252$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 252.72.1-63.a.1.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.72.1-63.a.2.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.72.1-63.b.1.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.72.1-63.b.2.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.72.1-63.c.1.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.72.1-63.c.2.8 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.192.7-63.a.1.11 | $252$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 252.504.19-63.a.1.16 | $252$ | $21$ | $21$ | $19$ | $?$ | not computed |