Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $324$ | ||
| Index: | $324$ | $\PSL_2$-index: | $162$ | ||||
| Genus: | $10 = 1 + \frac{ 162 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 9 }{2}$ | ||||||
| Cusps: | $9$ (none of which are rational) | Cusp widths | $18^{9}$ | Cusp orbits | $3^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18A10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.324.10.9 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}3&22\\10&19\end{bmatrix}$, $\begin{bmatrix}9&26\\22&29\end{bmatrix}$, $\begin{bmatrix}31&8\\14&5\end{bmatrix}$, $\begin{bmatrix}35&24\\8&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.162.10.a.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $24$ |
| Cyclic 36-torsion field degree: | $288$ |
| Full 36-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{36}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}$ |
| Newforms: | 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 162.2.a.c$^{2}$, 162.2.a.d$^{2}$, 324.2.a.b, 324.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ x w - x u + x v - x s - y t - y u + y r $ |
| $=$ | $x^{2} - x y - x t - x r + x s - y t + y u - y s - z r - w^{2} + w t + w s - t s + u^{2} - u v - u r + \cdots - r a$ | |
| $=$ | $x^{2} - x y + 2 x z + x w + x t - x v - 2 x s + x a + y z - 2 y w + y t - 2 y u - y v - y a + z v + \cdots - s^{2}$ | |
| $=$ | $x^{2} + x y + x z + 2 x w - x t + x u + x v - x r - x s + y^{2} + 2 y z - y t + y v - y r - y s + \cdots + s^{2}$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 18.81.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle v-r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+y-w$ |
| $\displaystyle W$ | $=$ | $\displaystyle x-y+w-t+v-r+s$ |
Equation of the image curve:
| $0$ | $=$ | $ 2XY+3XZ+YZ+2XW-YW-2ZW-W^{2} $ |
| $=$ | $ X^{3}+11X^{2}Y+17XY^{2}-3Y^{3}-7XYZ-5Y^{2}Z+8YZ^{2}-Z^{3}-4X^{2}W-6XYW-4Y^{2}W+2XZW-4Z^{2}W-2XW^{2}-2YW^{2}+2ZW^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $27$ | $27$ | $0$ | $0$ | full Jacobian |
| 36.108.2-18.a.1.1 | $36$ | $3$ | $3$ | $2$ | $0$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.648.22-36.a.1.3 | $36$ | $2$ | $2$ | $22$ | $6$ | $1^{10}\cdot2$ |
| 36.648.22-36.b.1.5 | $36$ | $2$ | $2$ | $22$ | $2$ | $1^{10}\cdot2$ |
| 36.648.22-36.c.1.4 | $36$ | $2$ | $2$ | $22$ | $5$ | $1^{10}\cdot2$ |
| 36.648.22-36.d.1.3 | $36$ | $2$ | $2$ | $22$ | $3$ | $1^{10}\cdot2$ |
| 36.972.28-18.a.1.8 | $36$ | $3$ | $3$ | $28$ | $3$ | $1^{18}$ |
| 36.1296.37-18.f.1.10 | $36$ | $4$ | $4$ | $37$ | $3$ | $1^{21}\cdot2^{3}$ |