Invariants
| Level: | $72$ | $\SL_2$-level: | $9$ | Newform level: | $27$ | ||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $3\cdot9$ | 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: | 9A1 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}21&13\\31&6\end{bmatrix}$, $\begin{bmatrix}37&69\\52&65\end{bmatrix}$, $\begin{bmatrix}55&14\\39&59\end{bmatrix}$, $\begin{bmatrix}70&33\\27&55\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.12.1.a.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $36$ |
| Cyclic 72-torsion field degree: | $864$ |
| Full 72-torsion field degree: | $248832$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 27.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{(y+z)^{2}(9y+z)^{3}}{z^{3}y(y+z)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.8.0-3.a.1.2 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 72.72.1-9.a.1.4 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.72.1-9.b.1.2 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.72.1-9.b.2.4 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.72.1-9.c.1.4 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.48.2-18.a.1.3 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-18.b.1.2 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.72.2-18.c.1.8 | $72$ | $3$ | $3$ | $2$ | $?$ | not computed |
| 72.48.2-36.a.1.4 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-36.b.1.2 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.96.4-36.c.1.7 | $72$ | $4$ | $4$ | $4$ | $?$ | not computed |
| 72.48.2-72.a.1.5 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.b.1.1 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.c.1.7 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 72.48.2-72.d.1.3 | $72$ | $2$ | $2$ | $2$ | $?$ | not computed |