Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $36$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $6$ are rational) | Cusp widths | $2^{9}\cdot18^{3}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18J1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.1.13 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}11&24\\0&7\end{bmatrix}$, $\begin{bmatrix}17&2\\0&13\end{bmatrix}$, $\begin{bmatrix}17&2\\18&5\end{bmatrix}$, $\begin{bmatrix}17&22\\0&11\end{bmatrix}$, $\begin{bmatrix}23&14\\0&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.72.1.a.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $2$ |
| Cyclic 36-torsion field degree: | $24$ |
| Full 36-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:-1:1)$, $(2:-3:1)$, $(0:1:0)$, $(0:1:1)$, $(-1:0:1)$, $(2:3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{2}+3z^{2})^{3}(y^{6}-15y^{4}z^{2}+75y^{2}z^{4}+3z^{6})^{3}}{z^{6}y^{2}(y-3z)^{2}(y-z)^{6}(y+z)^{6}(y+3z)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.a.1.9 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.72.0-18.a.1.1 | $36$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 36.72.0-18.a.1.12 | $36$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 36.288.3-36.a.1.6 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.3-36.a.1.11 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.3-36.a.1.16 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.3-36.a.2.7 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.3-36.a.2.9 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.3-36.a.2.16 | $36$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.5-36.a.1.2 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.a.1.12 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.b.1.7 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.b.1.10 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.c.1.5 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.c.1.9 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.d.1.4 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.d.1.10 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.7-36.a.1.4 | $36$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 36.288.7-36.a.2.1 | $36$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-18.d.1.5 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-18.d.2.8 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-18.e.1.7 | $36$ | $3$ | $3$ | $7$ | $0$ | $1^{6}$ |
| 36.432.10-18.a.1.7 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{9}$ |
| 72.288.3-72.a.1.14 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.a.1.18 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.a.1.29 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.a.2.12 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.a.2.18 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.a.2.25 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.5-72.a.1.9 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.a.1.15 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.b.1.9 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.b.1.15 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.c.1.11 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.c.1.13 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.d.1.11 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.d.1.13 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.7-72.a.1.5 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 72.288.7-72.a.2.1 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 108.432.7-54.a.1.5 | $108$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 108.432.10-54.a.1.7 | $108$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 108.432.13-54.a.1.3 | $108$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 180.288.3-180.a.1.3 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.3-180.a.1.30 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.3-180.a.1.32 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.3-180.a.2.7 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.3-180.a.2.26 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.3-180.a.2.30 | $180$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.5-180.a.1.4 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.a.1.20 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.b.1.9 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.b.1.18 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.c.1.10 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.c.1.19 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.d.1.3 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.d.1.20 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.7-180.a.1.10 | $180$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 180.288.7-180.a.2.1 | $180$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 252.288.3-252.a.1.4 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.3-252.a.1.30 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.3-252.a.1.31 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.3-252.a.2.8 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.3-252.a.2.26 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.3-252.a.2.29 | $252$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.5-252.a.1.12 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.a.1.16 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.b.1.4 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.b.1.20 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.c.1.7 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.c.1.20 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.d.1.10 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.d.1.17 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.7-252.a.1.7 | $252$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 252.288.7-252.a.2.1 | $252$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 252.432.7-126.d.1.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.d.2.15 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.e.1.15 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.e.2.15 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.f.1.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.f.2.15 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |