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: | 36G4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.160 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}17&1\\24&19\end{bmatrix}$, $\begin{bmatrix}19&14\\12&5\end{bmatrix}$, $\begin{bmatrix}25&31\\30&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.4.n.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: | $2^{8}\cdot3^{12}$ |
| 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 x^{2} - y z - z^{2} $ |
| $=$ | $3 x^{3} - x y^{2} + 7 x y z - x z^{2} - 3 w^{3}$ |
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 |
|---|
| $(0:1:0:0)$, $(0:-1:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^3}\cdot\frac{1222374888xyz^{7}w^{3}+1057536xyzw^{9}+1073705688xz^{8}w^{3}+9963000xz^{2}w^{9}+y^{12}-648y^{6}w^{6}-954396136y^{2}z^{10}-13204944y^{2}z^{4}w^{6}-954392768yz^{11}-53574264yz^{5}w^{6}+4096z^{12}-408059208z^{6}w^{6}-122472w^{12}}{w^{3}z^{5}(24xyz^{2}-yw^{3}-9zw^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 36.72.4.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
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.i.1.7 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.72.2-18.c.1.11 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 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 |
|---|---|---|---|---|---|---|
| 36.288.9-36.c.1.9 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.k.1.5 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 36.288.9-36.cf.1.4 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 36.288.9-36.ch.1.5 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.432.10-36.t.1.5 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.t.2.6 | $36$ | $3$ | $3$ | $10$ | $1$ | $2^{3}$ |
| 36.432.10-36.z.1.7 | $36$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 36.432.10-36.bd.1.5 | $36$ | $3$ | $3$ | $10$ | $5$ | $1^{6}$ |
| 72.288.9-72.m.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bf.1.2 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.ez.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.fg.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cp.1.5 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cq.1.5 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dg.1.7 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dh.1.5 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gg.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gh.1.6 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gy.1.6 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gz.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-252.bh.1.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bh.2.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bp.1.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.bp.2.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.ca.1.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-252.ca.2.9 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |