Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $432$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{2}\cdot12^{2}\cdot18^{2}\cdot36^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36C9 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}3&67\\40&15\end{bmatrix}$, $\begin{bmatrix}11&15\\0&41\end{bmatrix}$, $\begin{bmatrix}23&27\\20&31\end{bmatrix}$, $\begin{bmatrix}29&48\\60&41\end{bmatrix}$, $\begin{bmatrix}39&28\\68&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.144.9.x.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $6$ |
| Cyclic 72-torsion field degree: | $144$ |
| Full 72-torsion field degree: | $20736$ |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ t v - u^{2} $ |
| $=$ | $t r + u^{2} - r s$ | |
| $=$ | $z r + w v$ | |
| $=$ | $x r - w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} y^{4} - 27 x^{2} y^{6} - x^{2} z^{6} + 81 y^{2} z^{6} $ |
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 |
|---|
| $(0:0:0:0:-1/2:0:0:0:1)$, $(0:0:0:0:0:0:-1:1:0)$, $(0:0:0:0:0:0:1:1:0)$, $(0:0:0:0:1/4:0:0:0:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
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 36.144.9.x.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}Y^{4}-27X^{2}Y^{6}-X^{2}Z^{6}+81Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-12.l.1.4 | $24$ | $3$ | $3$ | $1$ | $0$ |
| 72.144.4-36.f.1.24 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.4-36.f.1.27 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.4-36.m.1.2 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.4-36.m.1.7 | $72$ | $2$ | $2$ | $4$ | $?$ |
| 72.144.5-36.m.1.6 | $72$ | $2$ | $2$ | $5$ | $?$ |
| 72.144.5-36.m.1.7 | $72$ | $2$ | $2$ | $5$ | $?$ |