Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $1^{6}\cdot4^{3}\cdot9^{2}\cdot36$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 36C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.1.44 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}7&30\\18&25\end{bmatrix}$, $\begin{bmatrix}11&31\\0&23\end{bmatrix}$, $\begin{bmatrix}23&26\\18&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 36.72.1.b.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $2$ |
| Cyclic 36-torsion field degree: | $24$ |
| Full 36-torsion field degree: | $2592$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 1 $ |
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.
| Weierstrass model |
|---|
| $(1:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{179280x^{2}y^{20}z^{2}+5378400x^{2}y^{18}z^{4}+4694351760x^{2}y^{16}z^{6}+102259221120x^{2}y^{14}z^{8}+1571021477280x^{2}y^{12}z^{10}+13615716824640x^{2}y^{10}z^{12}+68066708188320x^{2}y^{8}z^{14}+193187051464320x^{2}y^{6}z^{16}+297400038132240x^{2}y^{4}z^{18}+228767494082400x^{2}y^{2}z^{20}+68630248224720x^{2}z^{22}+720xy^{22}z+23760xy^{20}z^{3}+397282320xy^{18}z^{5}+9926355600xy^{16}z^{7}+276407056800xy^{14}z^{9}+3536474528160xy^{12}z^{11}+25869782604960xy^{10}z^{13}+110715996350880xy^{8}z^{15}+271128656192400xy^{6}z^{17}+366026842619280xy^{4}z^{19}+251644243490640xy^{2}z^{21}+68630248224720xz^{23}+y^{24}+36y^{22}z^{2}+16955082y^{20}z^{4}+474731604y^{18}z^{6}+39397958631y^{16}z^{8}+712884090312y^{14}z^{10}+7510184462700y^{12}z^{12}+46936113765192y^{10}z^{14}+173122262143839y^{8}z^{16}+369415652683284y^{6}z^{18}+442285170689898y^{4}z^{20}+274522542580836y^{2}z^{22}+68630635645209z^{24}}{zy^{4}(y^{2}+9z^{2})(24x^{2}y^{14}z+504x^{2}y^{12}z^{3}-1134x^{2}y^{10}z^{5}-51030x^{2}y^{8}z^{7}-288684x^{2}y^{6}z^{9}-708588x^{2}y^{4}z^{11}-826686x^{2}y^{2}z^{13}-354294x^{2}z^{15}-xy^{16}-24xy^{14}z^{2}+1260xy^{12}z^{4}+22680xy^{10}z^{6}+132678xy^{8}z^{8}+367416xy^{6}z^{10}+551124xy^{4}z^{12}+472392xy^{2}z^{14}+177147xz^{16}-252y^{14}z^{3}-4788y^{12}z^{5}-24948y^{10}z^{7}-20412y^{8}z^{9}+177147y^{6}z^{11}+531441y^{4}z^{13}+531441y^{2}z^{15}+177147z^{17})}$ |
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 |
| 18.72.0-18.a.1.3 | $18$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 36.72.0-18.a.1.12 | $36$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 36.288.5-36.a.1.2 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.e.1.4 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.i.1.3 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.j.1.4 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.432.7-36.t.1.4 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-36.t.2.8 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-36.v.1.8 | $36$ | $3$ | $3$ | $7$ | $0$ | $1^{6}$ |
| 36.432.10-36.f.1.8 | $36$ | $3$ | $3$ | $10$ | $2$ | $1^{9}$ |
| 72.288.3-72.d.1.7 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.d.2.14 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.e.1.8 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.3-72.e.2.16 | $72$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 72.288.5-72.e.1.12 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.n.1.12 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.y.1.12 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.bb.1.12 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.7-72.c.1.10 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 72.288.7-72.c.2.3 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 72.288.7-72.d.1.12 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 72.288.7-72.d.2.7 | $72$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 108.432.7-108.b.1.7 | $108$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 108.432.10-108.b.1.8 | $108$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 108.432.13-108.b.1.2 | $108$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 180.288.5-180.i.1.2 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.j.1.8 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.m.1.6 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.n.1.8 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.i.1.8 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.j.1.8 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.m.1.8 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.n.1.8 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.432.7-252.s.1.14 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.s.2.15 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.u.1.12 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.u.2.16 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.w.1.14 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.w.2.14 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |