Invariants
| Level: | $72$ | $\SL_2$-level: | $36$ | Newform level: | $432$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{3}\cdot18^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18C4 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}7&48\\58&65\end{bmatrix}$, $\begin{bmatrix}14&13\\3&4\end{bmatrix}$, $\begin{bmatrix}35&30\\44&1\end{bmatrix}$, $\begin{bmatrix}62&13\\27&22\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.m.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $288$ |
| Full 72-torsion field degree: | $41472$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x z - y w $ |
| $=$ | $3 x^{3} - x y^{2} + 3 z^{3} - 9 z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 3 x^{4} y^{2} - 3 x^{2} y z^{3} + y^{3} z^{3} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1)$, $(0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{12285x^{2}y^{10}+257985x^{2}y^{7}w^{3}+633015x^{2}y^{4}w^{6}-91125x^{2}yw^{9}+y^{12}-12285y^{9}z^{2}w+32796y^{9}w^{3}-157545y^{6}z^{2}w^{4}+456030y^{6}w^{6}-30375y^{3}z^{2}w^{7}+209196y^{3}w^{9}+995085z^{2}w^{10}+729w^{12}}{w^{4}y(9x^{2}y^{3}w^{2}+27x^{2}w^{5}+3y^{5}z^{2}-4y^{5}w^{2}+9y^{2}z^{2}w^{3}-28y^{2}w^{5})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.72.4.m.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-3X^{4}Y^{2}-3X^{2}YZ^{3}+Y^{3}Z^{3} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-12.h.1.6 | $24$ | $3$ | $3$ | $0$ | $0$ |
| 72.72.2-18.d.1.1 | $72$ | $2$ | $2$ | $2$ | $?$ |
| 72.72.2-18.d.1.4 | $72$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.288.9-36.t.1.1 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-36.x.1.3 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-36.cg.1.3 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-36.ck.1.4 | $72$ | $2$ | $2$ | $9$ |
| 72.432.10-36.s.1.2 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-36.s.2.2 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-36.ba.1.8 | $72$ | $3$ | $3$ | $10$ |
| 72.432.10-36.bc.1.3 | $72$ | $3$ | $3$ | $10$ |
| 72.288.9-72.ck.1.8 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.cw.1.4 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.fb.1.2 | $72$ | $2$ | $2$ | $9$ |
| 72.288.9-72.fn.1.1 | $72$ | $2$ | $2$ | $9$ |
| 216.432.16-108.s.1.2 | $216$ | $3$ | $3$ | $16$ |
| 216.432.16-108.t.1.2 | $216$ | $3$ | $3$ | $16$ |
| 216.432.16-108.u.1.2 | $216$ | $3$ | $3$ | $16$ |