Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $216$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6\cdot12\cdot18\cdot36$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36B5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.5.91 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}7&5\\34&3\end{bmatrix}$, $\begin{bmatrix}15&32\\14&3\end{bmatrix}$, $\begin{bmatrix}21&22\\10&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.5.h.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: | $2^{9}\cdot3^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 27.2.a.a$^{2}$, 72.2.a.a, 216.2.a.a, 216.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ x u^{2} + z t u $ |
| $=$ | $x z u + z^{2} t$ | |
| $=$ | $x y u + y z t$ | |
| $=$ | $x u v + z t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{11} - 4 x^{6} y z^{4} - 27 x^{5} z^{6} - 3 x y^{2} z^{8} + 27 y z^{10} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ -135x^{6} + 11664 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:0:1:0)$, $(0:1:0:1:0:0:0)$, $(0:1/2:0:1:0:0:0)$, $(0:-1:0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2}\cdot\frac{18144xu^{4}v^{2}+21760xuv^{5}-30618y^{2}w^{5}-23328yw^{6}-18656yv^{6}+2592zu^{5}v-11408zu^{2}v^{4}+18954w^{7}+12393w^{5}uv+19494w^{3}u^{2}v^{2}-29416wtuv^{4}+580wu^{3}v^{3}-18928wv^{6}}{v^{2}(19xuv^{3}+4yv^{4}-2zu^{2}v^{2}-27w^{3}u^{2}-25wtuv^{2}-2wu^{3}v-4wv^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 36.72.5.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{9}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{11}-4X^{6}YZ^{4}-27X^{5}Z^{6}-3XY^{2}Z^{8}+27YZ^{10} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 36.72.5.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 54z^{6}+\frac{9}{2}zwu^{4}-u^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.i.1.4 | $12$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 36.72.2-18.c.1.4 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
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.d.1.5 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.288.9-36.k.1.2 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 36.288.9-36.s.1.2 | $36$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 36.288.9-36.u.1.3 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.432.13-36.w.1.4 | $36$ | $3$ | $3$ | $13$ | $1$ | $1^{8}$ |
| 36.432.13-36.bv.1.3 | $36$ | $3$ | $3$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 36.432.13-36.bx.1.3 | $36$ | $3$ | $3$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 36.432.13-36.bx.2.4 | $36$ | $3$ | $3$ | $13$ | $1$ | $2^{2}\cdot4$ |
| 72.288.9-72.q.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bg.1.6 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.ci.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.cq.1.4 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.do.1.2 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dp.1.3 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.ds.1.6 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dt.1.4 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hh.1.7 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hi.1.6 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hl.1.4 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hm.1.7 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.13-252.ev.1.4 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.ev.2.4 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.ex.1.6 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.ex.2.4 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.ez.1.6 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.ez.2.4 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |