Invariants
| Level: | $36$ | $\SL_2$-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: | $1$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.142 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}17&11\\6&5\end{bmatrix}$, $\begin{bmatrix}23&8\\6&11\end{bmatrix}$, $\begin{bmatrix}23&12\\18&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.l.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $6$ |
| Cyclic 36-torsion field degree: | $36$ |
| Full 36-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 27.2.a.a$^{2}$, 432.2.a.d, 432.2.a.e |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 y^{2} + 4 z w - w^{2} $ |
| $=$ | $12 x^{3} + y z^{2} + 2 y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{5} - 10 x^{3} z^{2} + 3 x z^{4} + y^{3} z^{2} $ |
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 |
|---|
| $(0:0:1/4:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z-w)^{3}(z^{3}+3z^{2}w+3zw^{2}-w^{3})^{3}}{w^{3}z^{6}(z+2w)^{2}(4z-w)}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.72.4.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{5}-10X^{3}Z^{2}+Y^{3}Z^{2}+3XZ^{4} $ |
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.h.1.2 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.72.2-18.c.1.2 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.288.9-36.s.1.2 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 36.288.9-36.w.1.6 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.cf.1.1 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 36.288.9-36.cj.1.5 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.432.10-36.r.1.4 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.r.2.8 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.ba.1.6 | $36$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 36.432.10-36.bb.1.2 | $36$ | $3$ | $3$ | $10$ | $5$ | $1^{6}$ |
| 72.288.9-72.ch.1.6 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.ct.1.6 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.ey.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.fk.1.6 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cl.1.6 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cm.1.7 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dc.1.2 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dd.1.3 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gc.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gd.1.3 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gu.1.3 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gv.1.1 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-252.bf.1.14 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bf.2.8 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bn.1.14 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bn.2.8 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.by.1.8 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.by.2.8 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |