Invariants
| Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $108$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{3}\cdot18^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.144.4.80 |
Level structure
| $\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}7&8\\6&1\end{bmatrix}$, $\begin{bmatrix}7&33\\0&11\end{bmatrix}$, $\begin{bmatrix}11&9\\0&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 18.72.4.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: | $2^{2}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 27.2.a.a$^{3}$, 108.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 2 y^{2} + 3 y z + y w + z^{2} + 2 z w $ |
| $=$ | $12 x^{3} + y^{3} - y^{2} z + y^{2} w + y z^{2} - 2 y z w + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{5} z - 10 x^{4} z^{2} + 3 x^{3} y^{3} - 40 x^{3} z^{3} - 9 x^{2} y^{3} z - 50 x^{2} z^{4} + \cdots - 14 z^{6} $ |
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.
| Canonical model |
|---|
| $(0:0:-2:1)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^{14}}{3^3}\cdot\frac{125yz^{11}+2350yz^{10}w+21860yz^{9}w^{2}-213546yz^{8}w^{3}+458922yz^{7}w^{4}-505254yz^{6}w^{5}+355452yz^{5}w^{6}-167694yz^{4}w^{7}+52377yz^{3}w^{8}-10432yz^{2}w^{9}+1216yzw^{10}-64yw^{11}+125z^{12}+2225z^{11}w+19135z^{10}w^{2}-57707z^{9}w^{3}+141z^{8}w^{4}+169131z^{7}w^{5}-247419z^{6}w^{6}+186927z^{5}w^{7}-89934z^{4}w^{8}+28352z^{3}w^{9}-5632z^{2}w^{10}+640zw^{11}-32w^{12}}{4047yz^{11}-29337yz^{10}w+70701yz^{9}w^{2}-75371yz^{8}w^{3}+38870yz^{7}w^{4}-9338yz^{6}w^{5}+602yz^{5}w^{6}+106yz^{4}w^{7}-37yz^{3}w^{8}+19yz^{2}w^{9}-7yzw^{10}+yw^{11}+1999z^{12}-8808z^{11}w+1881z^{10}w^{2}+28690z^{9}w^{3}-40322z^{8}w^{4}+21848z^{7}w^{5}-5182z^{6}w^{6}+380z^{5}w^{7}-29z^{4}w^{8}+32z^{3}w^{9}-11z^{2}w^{10}+2zw^{11}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 18.72.4.k.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y+\frac{1}{2}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{3}Y^{3}-4X^{5}Z-9X^{2}Y^{3}Z-10X^{4}Z^{2}+9XY^{3}Z^{2}-40X^{3}Z^{3}-3Y^{3}Z^{3}-50X^{2}Z^{4}-44XZ^{5}-14Z^{6} $ |
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 |
| 36.72.2-18.c.1.3 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 36.72.2-18.c.1.7 | $36$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
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.q.1.4 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.u.1.3 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.cd.1.1 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.288.9-36.ch.1.4 | $36$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.432.10-18.o.1.3 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-18.o.2.6 | $36$ | $3$ | $3$ | $10$ | $0$ | $2^{3}$ |
| 36.432.10-18.u.1.5 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 36.432.10-18.v.1.3 | $36$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 72.288.9-72.cc.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.co.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.es.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 72.288.9-72.fe.1.7 | $72$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.ch.1.4 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cj.1.4 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.cy.1.2 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.288.9-180.da.1.2 | $180$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.fy.1.3 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.ga.1.7 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gq.1.3 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.288.9-252.gs.1.1 | $252$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 252.432.10-126.ch.1.10 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-126.ch.2.12 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-126.co.1.12 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-126.co.2.12 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-126.cw.1.10 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-126.cw.2.12 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |