Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $432$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 36.144.5.25 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&5\\0&31\end{bmatrix}$, $\begin{bmatrix}5&20\\0&1\end{bmatrix}$, $\begin{bmatrix}11&19\\30&1\end{bmatrix}$, $\begin{bmatrix}17&11\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.5.k.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: | $2^{12}\cdot3^{13}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 27.2.a.a$^{2}$, 48.2.a.a, 432.2.a.a, 432.2.a.h |
Models
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ - z t^{2} + w^{2} t $ |
| $=$ | $x t^{2} + w^{3}$ | |
| $=$ | $x t^{2} + z w t$ | |
| $=$ | $ - z t v + w^{2} v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{11} - 4 x^{6} y z^{4} - x^{5} z^{6} - 3 x y^{2} z^{8} + y z^{10} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ -5x^{6} + 16 $ |
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 |
|---|
| $(0:0:0:0:1:0:0)$, $(0:-1:0:0:0:1:0)$, $(0:1:0:0:0:1:0)$, $(0:1/2:0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{672xt^{4}v^{2}+492xtv^{5}+1818xu^{2}v^{4}+42y^{2}u^{5}+32yu^{6}-564yv^{6}-794zu^{3}v^{3}+96wt^{5}v+444wt^{2}v^{4}-361wu^{4}v^{2}+51tu^{5}v-26u^{7}+282uv^{6}}{v^{2}(3xu^{2}v^{2}-2yv^{4}+2zu^{3}v+wu^{4}-uv^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 36.72.5.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{11}-4X^{6}YZ^{4}-X^{5}Z^{6}-3XY^{2}Z^{8}+YZ^{10} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 36.72.5.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2w^{6}+\frac{3}{2}wt^{4}u-t^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 9.24.1-9.a.1.1 | $9$ | $6$ | $6$ | $1$ | $0$ | $1^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 18.72.2-18.c.1.3 | $18$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
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.a.1.11 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.288.9-36.n.1.9 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.288.9-36.ch.1.4 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.288.9-36.cj.1.5 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 36.288.10-36.y.1.4 | $36$ | $2$ | $2$ | $10$ | $1$ | $1^{5}$ |
| 36.288.10-36.z.1.4 | $36$ | $2$ | $2$ | $10$ | $1$ | $1^{5}$ |
| 36.288.10-36.ba.1.8 | $36$ | $2$ | $2$ | $10$ | $1$ | $1^{5}$ |
| 36.288.10-36.bb.1.8 | $36$ | $2$ | $2$ | $10$ | $1$ | $1^{5}$ |
| 36.432.13-36.bm.1.1 | $36$ | $3$ | $3$ | $13$ | $1$ | $1^{8}$ |
| 36.432.13-36.ca.1.12 | $36$ | $3$ | $3$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 36.432.13-36.ca.2.11 | $36$ | $3$ | $3$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 36.432.13-36.cc.1.12 | $36$ | $3$ | $3$ | $13$ | $6$ | $1^{4}\cdot2^{2}$ |
| 72.288.9-72.t.1.11 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.bp.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.fi.1.11 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.fm.1.10 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.10-72.be.1.16 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bf.1.16 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bg.1.32 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bh.1.32 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bi.1.12 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bj.1.14 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bk.1.24 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bl.1.28 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bm.1.32 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bn.1.32 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bo.1.16 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.10-72.bp.1.16 | $72$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 72.288.11-72.c.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.d.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.ce.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.cf.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.da.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.db.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.de.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 72.288.11-72.df.1.16 | $72$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 180.288.9-180.dy.1.10 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.dz.1.11 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.ec.1.8 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.ed.1.8 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.10-180.w.1.12 | $180$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 180.288.10-180.x.1.12 | $180$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 180.288.10-180.y.1.16 | $180$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 180.288.10-180.z.1.16 | $180$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 252.288.9-252.hr.1.11 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hs.1.10 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hv.1.11 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.hw.1.10 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.10-252.dy.1.14 | $252$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 252.288.10-252.dz.1.12 | $252$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 252.288.10-252.ea.1.16 | $252$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 252.288.10-252.eb.1.16 | $252$ | $2$ | $2$ | $10$ | $?$ | not computed |
| 252.432.13-252.fc.1.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.fc.2.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.fe.1.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.fe.2.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.fg.1.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 252.432.13-252.fg.2.24 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |