Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $27$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $3\cdot6\cdot9\cdot18$ | 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: | 18E2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.72.2.24 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}1&2\\12&35\end{bmatrix}$, $\begin{bmatrix}1&22\\30&19\end{bmatrix}$, $\begin{bmatrix}5&26\\30&7\end{bmatrix}$, $\begin{bmatrix}7&7\\24&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.36.2.c.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $6$ |
| Cyclic 36-torsion field degree: | $72$ |
| Full 36-torsion field degree: | $5184$ |
Jacobian
| Conductor: | $3^{6}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 27.2.a.a$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} z + x y^{2} - x w^{2} + y z^{2} $ |
| $=$ | $ - 2 x^{2} z + 2 x y^{2} - 2 x w^{2} + z^{2} w$ | |
| $=$ | $ - 2 x y z - x z w + 2 y^{3} - y^{2} w - 2 y w^{2} + w^{3}$ | |
| $=$ | $4 x^{2} y + 2 y^{2} z - y z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{5} - 4 x^{3} y z - x^{2} z^{3} - 3 x y^{2} z^{2} + y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{3} y $ | $=$ | $ 5x^{3} + 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 |
|---|
| $(0:0:1:0)$, $(0:-1:0:1)$, $(0:1:0:1)$, $(0:1/2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\cdot3^3\,\frac{384xyz^{4}w^{2}-492xyzw^{5}-192xz^{7}+24xz^{4}w^{3}-471xzw^{6}+192y^{2}z^{6}+520y^{2}z^{3}w^{3}-95y^{2}w^{6}-192yz^{6}w+504yz^{3}w^{4}-48yw^{7}+96z^{6}w^{2}-34z^{3}w^{5}+47w^{8}}{z^{3}(24xyzw^{2}+18xzw^{3}-8y^{2}z^{3}-2y^{2}w^{3}-12yz^{3}w+3yw^{4}-4z^{3}w^{2}-w^{5})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 18.36.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{5}-4X^{3}YZ-3XY^{2}Z^{2}-X^{2}Z^{3}+YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 18.36.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x^{3}+\frac{3}{2}xzw-z^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 36.144.4-18.a.1.8 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-18.k.1.3 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.216.4-18.c.1.12 | $36$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 36.216.4-18.e.1.6 | $36$ | $3$ | $3$ | $4$ | $0$ | $2$ |
| 36.216.4-18.e.2.8 | $36$ | $3$ | $3$ | $4$ | $0$ | $2$ |
| 36.216.4-18.f.1.11 | $36$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-36.a.1.8 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.c.1.5 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.d.1.11 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-36.l.1.2 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.n.1.1 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.o.1.3 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.5-36.h.1.6 | $36$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 36.144.5-36.i.1.8 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 36.144.5-36.j.1.2 | $36$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 36.144.5-36.k.1.8 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 72.144.4-72.a.1.4 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.c.1.4 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.e.1.11 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.f.1.10 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.q.1.10 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.s.1.6 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.u.1.7 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.v.1.4 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.5-72.ba.1.5 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bb.1.9 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bc.1.3 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bd.1.3 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.4-90.e.1.10 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-90.g.1.8 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.360.14-90.c.1.6 | $180$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 180.432.15-90.c.1.13 | $180$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 252.144.4-126.w.1.1 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-126.ba.1.1 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.216.4-126.d.1.10 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.d.2.12 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.e.1.12 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.e.2.10 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.f.1.12 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.f.2.10 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 180.144.4-180.e.1.10 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.f.1.6 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.g.1.8 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.i.1.10 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.j.1.2 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.k.1.4 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.5-180.n.1.13 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.o.1.14 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.p.1.9 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.q.1.10 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.4-252.p.1.8 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.q.1.4 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.r.1.4 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.v.1.6 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.w.1.3 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.x.1.3 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.5-252.n.1.6 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.o.1.6 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.p.1.5 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.q.1.5 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |