Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $108$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $3^{2}\cdot6\cdot9^{2}\cdot12^{2}\cdot18\cdot36^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36J8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.288.8.123 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&11\\24&1\end{bmatrix}$, $\begin{bmatrix}19&6\\0&25\end{bmatrix}$, $\begin{bmatrix}29&28\\12&31\end{bmatrix}$ |
| $\GL_2(\Z/36\Z)$-subgroup: | $C_6^2:C_6^2$ |
| Contains $-I$: | no $\quad$ (see 36.144.8.f.2 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $3$ |
| Cyclic 36-torsion field degree: | $36$ |
| Full 36-torsion field degree: | $1296$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{24}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot4$ |
| Newforms: | 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 108.2.b.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ y v - z u $ |
| $=$ | $x v - y v - w u$ | |
| $=$ | $x v + y t$ | |
| $=$ | $x u - x r - w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9 x^{8} y^{3} + x^{7} z^{4} - 18 x^{6} y^{3} z^{2} - x^{5} z^{6} - 8 x^{4} y^{3} z^{4} + \cdots + y^{3} z^{8} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:1:0:1)$, $(0:0:0:0:1:-1:1:1)$, $(0:0:0:0:-1:-1:-1:1)$, $(0:0:1:0:0:0:0:0)$ |
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 36.72.4.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
| $\displaystyle W$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ YZ+XW $ |
| $=$ | $ X^{3}-9XY^{2}-Z^{2}W+W^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.144.8.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ -9X^{8}Y^{3}-18X^{6}Y^{3}Z^{2}+X^{7}Z^{4}-8X^{4}Y^{3}Z^{4}-X^{5}Z^{6}+2X^{2}Y^{3}Z^{6}+Y^{3}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.c.3.3 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.144.4-36.f.1.3 | $36$ | $2$ | $2$ | $4$ | $0$ | $4$ |
| 36.144.4-36.f.1.8 | $36$ | $2$ | $2$ | $4$ | $0$ | $4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.576.17-36.b.3.2 | $36$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot4$ |
| 36.576.17-36.j.2.6 | $36$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot4$ |
| 36.576.17-36.k.2.3 | $36$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot4$ |
| 36.576.17-36.n.1.4 | $36$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot4$ |
| 36.864.22-36.o.4.5 | $36$ | $3$ | $3$ | $22$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 36.864.22-36.s.1.6 | $36$ | $3$ | $3$ | $22$ | $0$ | $2^{3}\cdot8$ |
| 36.864.22-36.t.1.4 | $36$ | $3$ | $3$ | $22$ | $0$ | $2^{3}\cdot8$ |
| 36.864.22-36.u.3.7 | $36$ | $3$ | $3$ | $22$ | $0$ | $1^{6}\cdot8$ |