Invariants
| Level: | $18$ | $\SL_2$-level: | $18$ | Newform level: | $324$ | ||
| Index: | $3888$ | $\PSL_2$-index: | $1944$ | ||||
| Genus: | $109 = 1 + \frac{ 1944 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 108 }{2}$ | ||||||
| Cusps: | $108$ (of which $9$ are rational) | Cusp widths | $18^{108}$ | Cusp orbits | $1^{9}\cdot2^{9}\cdot3^{3}\cdot6^{12}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $20 \le \gamma \le 36$ | ||||||
| $\overline{\Q}$-gonality: | $20 \le \gamma \le 36$ | ||||||
| Rational cusps: | $9$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 18.3888.109.1 |
Level structure
| $\GL_2(\Z/18\Z)$-generators: | $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/18\Z)$-subgroup: | $C_6$ |
| Contains $-I$: | no $\quad$ (see 18.1944.109.a.1 for the level structure with $-I$) |
| Cyclic 18-isogeny field degree: | $1$ |
| Cyclic 18-torsion field degree: | $1$ |
| Full 18-torsion field degree: | $6$ |
Jacobian
| Conductor: | $2^{106}\cdot3^{366}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{31}\cdot2^{33}\cdot4^{3}$ |
| Newforms: | 18.2.c.a$^{6}$, 27.2.a.a$^{6}$, 36.2.a.a$^{3}$, 36.2.e.a$^{3}$, 54.2.a.a$^{4}$, 54.2.a.b$^{4}$, 54.2.c.a$^{4}$, 81.2.a.a$^{3}$, 81.2.c.a$^{3}$, 81.2.c.b$^{3}$, 108.2.a.a$^{2}$, 108.2.e.a$^{2}$, 162.2.a.a$^{2}$, 162.2.a.b$^{2}$, 162.2.a.c$^{2}$, 162.2.a.d$^{2}$, 162.2.c.a$^{2}$, 162.2.c.b$^{2}$, 162.2.c.c$^{2}$, 162.2.c.d$^{2}$, 324.2.a.a, 324.2.a.b, 324.2.a.c, 324.2.a.d, 324.2.e.a, 324.2.e.b, 324.2.e.c, 324.2.e.d |
Rational points
This modular curve has 9 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $648$ | $324$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{arith}}(9)$ | $9$ | $6$ | $6$ | $10$ | $0$ | $1^{29}\cdot2^{31}\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 18.1296.28-18.a.1.1 | $18$ | $3$ | $3$ | $28$ | $0$ | $1^{21}\cdot2^{24}\cdot4^{3}$ |
| 18.1296.28-18.a.2.1 | $18$ | $3$ | $3$ | $28$ | $0$ | $1^{21}\cdot2^{24}\cdot4^{3}$ |
| 18.1296.37-18.a.1.1 | $18$ | $3$ | $3$ | $37$ | $2$ | $1^{24}\cdot2^{18}\cdot4^{3}$ |
| 18.1296.37-18.b.1.1 | $18$ | $3$ | $3$ | $37$ | $1$ | $1^{24}\cdot2^{24}$ |
| 18.1296.37-18.c.1.1 | $18$ | $3$ | $3$ | $37$ | $3$ | $2^{30}\cdot4^{3}$ |
| 18.1296.37-18.h.1.1 | $18$ | $3$ | $3$ | $37$ | $1$ | $1^{20}\cdot2^{22}\cdot4^{2}$ |
| 18.1296.37-18.k.1.1 | $18$ | $3$ | $3$ | $37$ | $1$ | $1^{20}\cdot2^{22}\cdot4^{2}$ |
| 18.1944.46-18.a.1.1 | $18$ | $2$ | $2$ | $46$ | $1$ | $1^{19}\cdot2^{20}\cdot4$ |
| 18.1944.46-18.a.1.4 | $18$ | $2$ | $2$ | $46$ | $1$ | $1^{19}\cdot2^{20}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 36.7776.235-36.a.1.2 | $36$ | $2$ | $2$ | $235$ | $3$ | $2^{3}\cdot4^{8}\cdot8^{9}\cdot16$ |
| 36.7776.235-36.b.1.1 | $36$ | $2$ | $2$ | $235$ | $3$ | $2^{3}\cdot4^{8}\cdot8^{9}\cdot16$ |
| 36.7776.253-36.a.1.2 | $36$ | $2$ | $2$ | $253$ | $24$ | $1^{38}\cdot2^{37}\cdot4^{8}$ |
| 36.7776.253-36.b.1.2 | $36$ | $2$ | $2$ | $253$ | $12$ | $1^{38}\cdot2^{37}\cdot4^{8}$ |
| 36.7776.253-36.c.1.2 | $36$ | $2$ | $2$ | $253$ | $12$ | $1^{38}\cdot2^{37}\cdot4^{8}$ |
| 36.7776.253-36.d.1.2 | $36$ | $2$ | $2$ | $253$ | $24$ | $1^{38}\cdot2^{37}\cdot4^{8}$ |
| 36.7776.271-36.k.1.4 | $36$ | $2$ | $2$ | $271$ | $15$ | $2^{41}\cdot4^{12}\cdot8^{4}$ |
| 36.7776.271-36.l.1.2 | $36$ | $2$ | $2$ | $271$ | $13$ | $2^{41}\cdot4^{12}\cdot8^{4}$ |
| 54.11664.379-54.a.1.1 | $54$ | $3$ | $3$ | $379$ | $3$ | $6^{6}\cdot12^{15}\cdot18^{3}$ |
| 54.11664.379-54.e.1.2 | $54$ | $3$ | $3$ | $379$ | $3$ | $6^{6}\cdot12^{15}\cdot18^{3}$ |
| 54.11664.379-54.g.1.1 | $54$ | $3$ | $3$ | $379$ | $42$ | $3^{4}\cdot6^{14}\cdot9^{2}\cdot12^{10}\cdot18^{2}$ |
| 54.11664.379-54.h.1.1 | $54$ | $3$ | $3$ | $379$ | $3$ | $6^{6}\cdot12^{15}\cdot18^{3}$ |
| 54.11664.406-54.g.1.1 | $54$ | $3$ | $3$ | $406$ | $35$ | $1^{43}\cdot2^{53}\cdot3^{12}\cdot4^{10}\cdot6^{12}$ |
| 54.11664.406-54.g.2.1 | $54$ | $3$ | $3$ | $406$ | $35$ | $1^{43}\cdot2^{53}\cdot3^{12}\cdot4^{10}\cdot6^{12}$ |
| 54.11664.433-54.c.1.1 | $54$ | $3$ | $3$ | $433$ | $45$ | $3^{6}\cdot6^{19}\cdot12^{14}\cdot24$ |