Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $108$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $2 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot18$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18B2 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}6&53\\59&0\end{bmatrix}$, $\begin{bmatrix}34&3\\55&14\end{bmatrix}$, $\begin{bmatrix}49&26\\57&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.24.2.b.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $124416$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x w^{2} + y z w $ |
| $=$ | $3 x z w + y z^{2}$ | |
| $=$ | $3 x y w + y^{2} z$ | |
| $=$ | $3 x^{2} w + x y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{3} y^{2} - x^{3} z^{2} + y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{3} y $ | $=$ | $ 1 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(1:0:0)$ | $(0:-1:1)$, $(0:1:1)$ | $(-1:1:0:0)$, $(1:1:0:0)$ | |
| no | $\infty$ | $0.000$ | |||||
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -3^3\,\frac{(z-w)^{3}(z+w)^{3}}{w^{3}y^{2}x}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 18.24.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y^{2}-X^{3}Z^{2}+YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 18.24.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}y^{3}zw^{2}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -yw$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.16.0-6.b.1.3 | $24$ | $3$ | $3$ | $0$ | $0$ |
| 72.24.1-9.a.1.4 | $72$ | $2$ | $2$ | $1$ | $?$ |
| 72.24.1-9.a.1.6 | $72$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 72.144.4-18.j.1.2 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-18.j.2.4 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-18.k.1.6 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-18.m.1.4 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-18.o.1.2 | $72$ | $3$ | $3$ | $4$ |
| 72.144.4-18.q.1.3 | $72$ | $3$ | $3$ | $4$ |
| 72.192.7-36.c.1.2 | $72$ | $4$ | $4$ | $7$ |