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 | $3^{2}\cdot9^{2}\cdot12\cdot36$ | 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: | 36F4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.103 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}1&16\\6&19\end{bmatrix}$, $\begin{bmatrix}19&27\\0&35\end{bmatrix}$, $\begin{bmatrix}29&32\\24&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.e.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $6$ |
| Cyclic 36-torsion field degree: | $72$ |
| Full 36-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}$ |
| Newforms: | 54.2.a.a, 54.2.a.b, 432.2.a.b, 432.2.a.g |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ - x w + y z $ |
| $=$ | $x^{3} + 9 x y^{2} - z^{2} w - w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{3} y^{3} - x^{2} z^{4} + 9 x y^{3} z^{2} - z^{6} $ |
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 |
|---|
| $(0:0:1:0)$, $(0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{22626x^{2}y^{4}w^{6}-6804xy^{8}w^{3}+275076xy^{2}w^{9}+729y^{12}+756y^{6}w^{6}+z^{12}+36z^{8}w^{4}-288z^{6}w^{6}+2862z^{4}w^{8}-27648z^{2}w^{10}-30834w^{12}}{w^{3}y^{4}(x^{2}w^{3}-9xy^{4}+y^{2}w^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.72.4.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y^{3}+9XY^{3}Z^{2}-X^{2}Z^{4}-Z^{6} $ |
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.f.1.3 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 18.72.2-18.d.1.3 | $18$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 36.72.2-18.d.1.1 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
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.e.1.12 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.288.9-36.j.1.5 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.288.9-36.r.1.4 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.288.9-36.t.1.3 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.432.10-36.f.1.8 | $36$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 36.432.10-36.l.1.5 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.l.2.4 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.n.1.1 | $36$ | $3$ | $3$ | $10$ | $1$ | $1^{6}$ |
| 72.288.8-72.e.1.2 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.e.2.16 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.f.1.4 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.f.2.15 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.9-72.k.1.6 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bc.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cb.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cj.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.10-72.i.1.13 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.i.2.2 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.j.1.15 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.j.2.1 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 108.432.16-108.d.1.8 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 108.432.16-108.f.1.5 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 108.432.16-108.h.1.1 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 180.288.9-180.y.1.12 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.z.1.4 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.bc.1.7 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.bd.1.6 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.y.1.6 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.z.1.7 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.bc.1.6 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.bd.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-252.m.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.m.2.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.o.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.o.2.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.q.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.q.2.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |