Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $36$ | ||
| 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 | $2^{9}\cdot18^{3}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| 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: | 18J1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.1.32 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&30\\18&29\end{bmatrix}$, $\begin{bmatrix}7&33\\0&35\end{bmatrix}$, $\begin{bmatrix}31&32\\18&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.72.1.d.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: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 27 $ |
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 |
|---|
| $(0:1:0)$, $(3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{2160x^{2}y^{22}-32152924080x^{2}y^{20}z^{2}+383969367833040x^{2}y^{18}z^{4}-280376631469034640x^{2}y^{16}z^{6}+10228781066963729760x^{2}y^{14}z^{8}+1441916433380984954400x^{2}y^{12}z^{10}+16783619179314218747040x^{2}y^{10}z^{12}-2117657858827373145015840x^{2}y^{8}z^{14}-167436557451643575680410320x^{2}y^{6}z^{16}-6817266959407137591123746160x^{2}y^{4}z^{18}-141309976344689905273404301680x^{2}y^{2}z^{20}-1144559119736874419328183385680x^{2}z^{22}-1613520xy^{22}z+1124347022640xy^{20}z^{3}-4465839321030960xy^{18}z^{5}+1444202698696664400xy^{16}z^{7}+43383401092986685920xy^{14}z^{9}-3684846910474873596960xy^{12}z^{11}-246456330021234908281440xy^{10}z^{13}-12473376516202440712099680xy^{8}z^{15}-694588176758305799559617040xy^{6}z^{17}-25096453142952371481029817360xy^{4}z^{19}-466263575703827494254895854960xy^{2}z^{21}-3433677359210623257984550157040xz^{23}-y^{24}+457772148y^{22}z^{2}-24691781154618y^{20}z^{4}+39733659638719668y^{18}z^{6}-4862698783242508791y^{16}z^{8}-464676307201930272984y^{14}z^{10}-9048119905742139405900y^{12}z^{12}-251360958212346834027576y^{10}z^{14}-41883091314716953742832639y^{8}z^{16}-2764844097998576475585006108y^{6}z^{18}-91011543978448970898851604762y^{4}z^{20}-1526144872930700209542411377148y^{2}z^{22}-10301090227368872814013340861289z^{24}}{y^{2}(y^{2}+243z^{2})^{2}(x^{2}y^{16}+40176x^{2}y^{14}z^{2}+734832x^{2}y^{12}z^{4}+982102968x^{2}y^{10}z^{6}-178988265918x^{2}y^{8}z^{8}+427033459159272x^{2}y^{4}z^{12}+4941387170271576x^{2}y^{2}z^{14}-50031545098999707x^{2}z^{16}+72xy^{16}z+420390xy^{14}z^{3}-64520874xy^{12}z^{5}+8025821982xy^{10}z^{7}+104603532030xy^{8}z^{9}-40104994180302xy^{6}z^{11}-686303773648830xy^{4}z^{13}+18530201888518410xy^{2}z^{15}+300189270593998242xz^{17}+2268y^{16}z^{2}+2326968y^{14}z^{4}-238026519y^{12}z^{6}-13114900998y^{10}z^{8}+2813835011607y^{8}z^{10}+64393934317668y^{6}z^{12}-3500149245609033y^{4}z^{14}-100063090197999414y^{2}z^{16}-450283905890997363z^{18})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.b.1.2 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 36.72.0-18.a.1.4 | $36$ | $2$ | $2$ | $0$ | $0$ | 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.i.1.3 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.k.1.6 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.m.1.3 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.288.5-36.o.1.1 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.432.7-18.l.1.2 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-18.l.2.5 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{3}$ |
| 36.432.7-18.p.1.5 | $36$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 36.432.10-18.u.1.5 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{9}$ |
| 72.288.5-72.z.1.6 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.bf.1.6 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.bs.1.7 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.288.5-72.by.1.7 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 108.432.7-54.c.1.5 | $108$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 108.432.10-54.n.1.3 | $108$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 108.432.13-54.r.1.3 | $108$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 180.288.5-180.q.1.6 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.r.1.6 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.y.1.2 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.z.1.2 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.q.1.8 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.r.1.6 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.y.1.2 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.288.5-252.z.1.4 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.432.7-126.cm.1.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.cm.2.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.cq.1.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.cq.2.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.cw.1.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-126.cw.2.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |