Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $54$ | ||
| 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: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 18D2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.72.2.9 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}13&5\\24&29\end{bmatrix}$, $\begin{bmatrix}13&30\\0&11\end{bmatrix}$, $\begin{bmatrix}17&9\\18&5\end{bmatrix}$, $\begin{bmatrix}35&11\\30&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.36.2.d.1 for the level structure with $-I$) |
| Cyclic 36-isogeny field degree: | $6$ |
| Cyclic 36-torsion field degree: | $36$ |
| Full 36-torsion field degree: | $5184$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 54.2.a.a, 54.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y w - x z w - w^{3} $ |
| $=$ | $x y z - x z^{2} - z w^{2}$ | |
| $=$ | $x y^{2} - x y z - y w^{2}$ | |
| $=$ | $x^{2} y - x^{2} z - x w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y + x^{3} z^{2} + 12 x^{2} y^{2} z + 7 x y z^{3} + z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + 1\right) y $ | $=$ | $ -9x^{3} $ |
Rational points
This modular curve has 4 rational cusps and 2 rational CM points, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(-1:0:1)$ | $(-1:-3:1)$ | $(-1:-1:0:1)$ | |
| no | $\infty$ | $0.000$ | |||||
| 36.a1 | $-12$ | $54000$ | $= 2^{4} \cdot 3^{3} \cdot 5^{3}$ | $10.897$ | $(-1:1/2:1)$ | $(-1:3:1)$ | $(-1:0:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{192x^{7}w-192x^{4}w^{4}+327xz^{6}w+7248xz^{3}w^{4}-2688xw^{7}+53y^{2}z^{6}-4096y^{2}z^{3}w^{3}-2304y^{2}w^{6}-16yz^{7}-6424yz^{4}w^{3}-576yzw^{6}-21z^{8}-1908z^{5}w^{3}+7200z^{2}w^{6}}{w(12xz^{6}-21xz^{3}w^{3}+22y^{2}z^{3}w^{2}-8y^{2}w^{5}+37yz^{4}w^{2}-20yzw^{5}+21z^{5}w^{2}-30z^{2}w^{5})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 18.36.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y+12X^{2}Y^{2}Z+X^{3}Z^{2}+7XYZ^{3}+Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 18.36.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x^{3}+6xzw+4w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
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.1.11 | $12$ | $3$ | $3$ | $0$ | $0$ | 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.b.1.3 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-36.b.1.3 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.e.1.6 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.f.1.12 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-18.l.1.2 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.4-36.m.1.2 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.p.1.2 | $36$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 36.144.4-36.q.1.2 | $36$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.144.5-36.l.1.5 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 36.144.5-36.m.1.5 | $36$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 36.144.5-36.n.1.1 | $36$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 36.144.5-36.o.1.5 | $36$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 36.216.4-18.c.1.8 | $36$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 36.216.4-18.g.1.1 | $36$ | $3$ | $3$ | $4$ | $0$ | $2$ |
| 36.216.4-18.g.2.3 | $36$ | $3$ | $3$ | $4$ | $0$ | $2$ |
| 36.216.4-18.h.1.1 | $36$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 72.144.4-72.b.1.8 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.d.1.10 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.g.1.9 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.h.1.11 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.r.1.14 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.t.1.10 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.w.1.3 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.4-72.x.1.5 | $72$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 72.144.5-72.be.1.10 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bf.1.14 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bg.1.10 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 72.144.5-72.bh.1.8 | $72$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 108.216.8-54.a.1.6 | $108$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 108.216.8-54.b.1.3 | $108$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 108.216.8-54.c.1.3 | $108$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 180.144.4-90.i.1.6 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-90.j.1.5 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.m.1.7 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.n.1.5 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.o.1.7 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.p.1.8 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.q.1.13 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.4-180.r.1.9 | $180$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 180.144.5-180.r.1.7 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.s.1.5 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.t.1.9 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.144.5-180.u.1.13 | $180$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.360.14-90.d.1.3 | $180$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 180.432.15-90.d.1.14 | $180$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 252.144.4-252.ba.1.2 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.bb.1.4 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.bc.1.4 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-126.bd.1.10 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.bd.1.10 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-126.be.1.9 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.be.1.6 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.4-252.bf.1.6 | $252$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 252.144.5-252.r.1.3 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.s.1.3 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.t.1.5 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.144.5-252.u.1.5 | $252$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.216.4-126.g.1.10 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.g.2.10 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.h.1.12 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.h.2.12 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.i.1.24 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 252.216.4-126.i.2.16 | $252$ | $3$ | $3$ | $4$ | $?$ | not computed |