Invariants
Level: | $36$ | $\SL_2$-level: | $36$ | Newform level: | $432$ | ||
Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{4}\cdot18^{4}\cdot36^{4}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $2$ | ||||||
$\Q$-gonality: | $6$ | ||||||
$\overline{\Q}$-gonality: | $6$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 36A17 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 36.576.17.125 |
Level structure
$\GL_2(\Z/36\Z)$-generators: | $\begin{bmatrix}5&27\\0&1\end{bmatrix}$, $\begin{bmatrix}29&2\\24&19\end{bmatrix}$ |
$\GL_2(\Z/36\Z)$-subgroup: | $C_3\times C_6^2:C_6$ |
Contains $-I$: | no $\quad$ (see 36.288.17.n.1 for the level structure with $-I$) |
Cyclic 36-isogeny field degree: | $3$ |
Cyclic 36-torsion field degree: | $18$ |
Full 36-torsion field degree: | $648$ |
Jacobian
Conductor: | $2^{40}\cdot3^{50}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{9}\cdot4^{2}$ |
Newforms: | 54.2.a.a$^{2}$, 54.2.a.b$^{2}$, 108.2.b.a$^{2}$, 144.2.a.b, 432.2.a.b, 432.2.a.c, 432.2.a.f, 432.2.a.g |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.192.1-12.g.1.2 | $12$ | $3$ | $3$ | $1$ | $0$ | $1^{8}\cdot4^{2}$ |
36.288.8-36.f.2.1 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
36.288.8-36.f.2.4 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
36.288.8-36.f.3.2 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
36.288.8-36.f.3.8 | $36$ | $2$ | $2$ | $8$ | $0$ | $1^{5}\cdot4$ |
36.288.9-36.x.1.1 | $36$ | $2$ | $2$ | $9$ | $2$ | $4^{2}$ |
36.288.9-36.x.1.4 | $36$ | $2$ | $2$ | $9$ | $2$ | $4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
36.1728.49-36.fj.2.4 | $36$ | $3$ | $3$ | $49$ | $2$ | $2^{6}\cdot4\cdot8^{2}$ |
36.1728.49-36.fj.3.2 | $36$ | $3$ | $3$ | $49$ | $2$ | $2^{6}\cdot4\cdot8^{2}$ |
36.1728.49-36.fk.1.1 | $36$ | $3$ | $3$ | $49$ | $3$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
36.1728.49-36.fp.2.3 | $36$ | $3$ | $3$ | $49$ | $6$ | $1^{12}\cdot2^{2}\cdot8^{2}$ |