Invariants
| Level: | $72$ | $\SL_2$-level: | $18$ | Newform level: | $27$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{3}\cdot9^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 9C1 |
Level structure
| $\GL_2(\Z/72\Z)$-generators: | $\begin{bmatrix}25&3\\8&35\end{bmatrix}$, $\begin{bmatrix}47&18\\57&5\end{bmatrix}$, $\begin{bmatrix}59&6\\49&55\end{bmatrix}$, $\begin{bmatrix}70&63\\15&52\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.36.1.a.1 for the level structure with $-I$) |
| Cyclic 72-isogeny field degree: | $12$ |
| Cyclic 72-torsion field degree: | $288$ |
| Full 72-torsion field degree: | $82944$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 27.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} - 7 $ |
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 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y+5z)^{12}(y^{3}+231y^{2}z+291yz^{2}+1637z^{3})^{3}}{z(y-4z)^{9}(y+5z)^{9}(y^{2}+yz+7z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.24.0-3.a.1.4 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 72.24.0-9.a.1.7 | $72$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 72.24.0-9.b.1.7 | $72$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 72.24.1-9.a.1.7 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 72.144.4-18.c.1.1 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-36.h.1.3 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.i.1.1 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.j.1.6 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-18.m.1.3 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-36.r.1.1 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.y.1.6 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.z.1.7 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.216.1-9.a.1.7 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.216.1-9.a.2.5 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.216.1-9.b.1.7 | $72$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 72.216.4-9.c.1.3 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.216.4-18.c.1.4 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.288.10-36.w.1.7 | $72$ | $4$ | $4$ | $10$ | $?$ | not computed |
| 216.216.4-27.a.1.6 | $216$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 216.216.7-27.a.1.8 | $216$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 216.216.7-27.b.1.8 | $216$ | $3$ | $3$ | $7$ | $?$ | not computed |