Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $54$ | ||
| 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: | 36F4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.111 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&35\\24&31\end{bmatrix}$, $\begin{bmatrix}19&16\\12&35\end{bmatrix}$, $\begin{bmatrix}35&17\\24&31\end{bmatrix}$, $\begin{bmatrix}35&28\\24&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.f.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: | $2^{4}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 54.2.a.a$^{2}$, 54.2.a.b$^{2}$ |
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 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 |
|---|
| $(0:0:1:0)$, $(3:1:0:0)$, $(0:0:-1:1)$, $(0:1:0:0)$, $(0:0:1:1)$, $(-3:1:0:0)$ |
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.f.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.g.1.3 | $12$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 36.288.8-36.f.1.4 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.1.5 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.2.4 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.2.5 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.3.2 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.3.7 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.4.3 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.8-36.f.4.6 | $36$ | $2$ | $2$ | $8$ | $0$ | $4$ |
| 36.288.9-36.f.1.4 | $36$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 36.288.9-36.p.1.3 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.288.9-36.v.1.2 | $36$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 36.288.9-36.x.1.1 | $36$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.432.10-36.g.1.5 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 36.432.10-36.m.1.6 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-36.m.2.3 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-36.o.1.8 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 72.288.8-72.g.1.8 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.g.1.25 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.g.2.10 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.g.2.23 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.h.1.15 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.h.1.18 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.h.2.4 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.h.2.29 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.1.11 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.1.22 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.2.6 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.2.27 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.3.12 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.3.21 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.4.10 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.8-72.n.4.23 | $72$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 72.288.9-72.l.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bu.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cn.1.3 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cv.1.3 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dg.1.8 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dg.1.25 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dh.1.30 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dh.1.33 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.di.1.3 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.di.1.30 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dj.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dj.1.29 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dk.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dk.1.29 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dl.1.3 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dl.1.30 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dm.1.16 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dm.1.17 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dn.1.8 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.dn.1.25 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.10-72.k.1.15 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.k.1.18 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.k.2.4 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.k.2.29 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.l.1.8 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.l.1.25 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.l.2.10 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.l.2.23 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 108.432.16-108.e.1.2 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 108.432.16-108.g.1.6 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 108.432.16-108.i.1.3 | $108$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 180.288.8-180.f.1.7 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.1.10 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.2.4 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.2.13 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.3.5 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.3.12 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.4.3 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.8-180.f.4.14 | $180$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 180.288.9-180.ba.1.6 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.bb.1.9 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.be.1.7 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.bf.1.3 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.8-252.f.1.7 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.1.10 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.2.4 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.2.13 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.3.7 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.3.10 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.4.4 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.8-252.f.4.13 | $252$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 252.288.9-252.ba.1.5 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.bb.1.5 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.be.1.3 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.bf.1.5 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-252.n.1.15 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.n.2.16 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.p.1.15 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.p.2.16 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.r.1.6 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.r.2.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |