Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $108$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $3^{2}\cdot9^{2}\cdot12\cdot36$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36G4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.124 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}1&35\\24&31\end{bmatrix}$, $\begin{bmatrix}23&25\\12&23\end{bmatrix}$, $\begin{bmatrix}25&26\\12&23\end{bmatrix}$, $\begin{bmatrix}31&33\\0&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.d.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $3$ |
| Cyclic 36-torsion field degree: | $36$ |
| Full 36-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 27.2.a.a$^{3}$, 108.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $x^{2} z - 5 x y z + 3 y^{2} z - 3 z^{3} - w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + x^{2} y^{3} + 10 x^{2} z^{3} - 9 z^{5} $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1:0:0:0)$, $(-8/3:-1/3:1:0)$, $(0:-1:1:0)$, $(-1:1:0:0)$, $(0:1:1:0)$, $(8/3:1/3:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^3}\cdot\frac{2187x^{12}-52488x^{6}w^{6}+1592136y^{12}-25920y^{6}w^{6}+326418448y^{2}z^{10}+249533800y^{2}z^{7}w^{3}+52919552y^{2}z^{4}w^{6}+3285728y^{2}zw^{9}-319052632z^{12}-288170976z^{9}w^{3}-83434904z^{6}w^{6}-9517824z^{3}w^{9}-367416w^{12}}{w^{3}z(8y^{2}z^{6}-2y^{2}z^{3}w^{3}-y^{2}w^{6}-8z^{8}+z^{5}w^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.72.4.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{3}-X^{4}Z+10X^{2}Z^{3}-9Z^{5} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.g.1.11 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 36.72.2-18.c.1.12 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.288.8-36.e.1.7 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.e.2.6 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.e.3.5 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.e.4.2 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.9-36.b.1.10 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.n.1.9 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.u.1.3 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.w.1.6 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.432.10-36.g.1.11 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 36.432.10-36.i.1.9 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-36.i.2.10 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-36.k.1.9 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 72.288.8-72.k.1.14 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.k.2.28 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.l.1.10 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.l.2.20 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.m.1.22 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.m.2.20 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.m.3.10 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.m.4.4 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.9-72.j.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bo.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cm.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cs.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cy.1.20 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cz.1.39 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.da.1.8 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.db.1.15 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dc.1.15 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dd.1.8 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.de.1.23 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.df.1.20 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.10-72.g.1.24 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.g.2.15 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.h.1.20 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.h.2.7 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 180.288.8-180.e.1.7 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.e.2.7 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.e.3.7 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.e.4.7 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.9-180.s.1.5 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.t.1.9 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.w.1.5 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.x.1.11 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.8-252.e.1.11 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.e.2.9 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.e.3.6 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.e.4.2 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.9-252.s.1.10 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.t.1.5 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.w.1.8 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.x.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-252.h.1.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.h.2.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.j.1.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.j.2.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.l.1.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.l.2.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |