Invariants
| Level: | $126$ | $\SL_2$-level: | $18$ | Newform level: | $1$ | ||
| Index: | $216$ | $\PSL_2$-index: | $108$ | ||||
| Genus: | $2 = 1 + \frac{ 108 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $3^{6}\cdot6^{6}\cdot9^{2}\cdot18^{2}$ | Cusp orbits | $2^{2}\cdot3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 18P2 |
Level structure
| $\GL_2(\Z/126\Z)$-generators: | $\begin{bmatrix}29&24\\30&1\end{bmatrix}$, $\begin{bmatrix}29&105\\36&89\end{bmatrix}$, $\begin{bmatrix}97&120\\48&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 126.108.2.c.1 for the level structure with $-I$) |
| Cyclic 126-isogeny field degree: | $24$ |
| Cyclic 126-torsion field degree: | $864$ |
| Full 126-torsion field degree: | $217728$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $72$ | $36$ | $0$ | $0$ |
| 63.72.0-63.c.1.2 | $63$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 42.72.0-6.a.1.1 | $42$ | $3$ | $3$ | $0$ | $0$ |
| 63.72.0-63.c.1.2 | $63$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 126.432.7-126.c.1.3 | $126$ | $2$ | $2$ | $7$ |
| 126.432.7-126.cf.1.2 | $126$ | $2$ | $2$ | $7$ |
| 126.432.7-126.cu.1.3 | $126$ | $2$ | $2$ | $7$ |
| 126.432.7-126.cv.1.2 | $126$ | $2$ | $2$ | $7$ |
| 252.432.7-252.e.1.9 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.q.1.3 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.r.1.2 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.ca.1.9 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.cb.1.3 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.cc.1.1 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.ed.1.3 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.ee.1.1 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.ef.1.9 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.eg.1.3 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.eh.1.1 | $252$ | $2$ | $2$ | $7$ |
| 252.432.7-252.ei.1.9 | $252$ | $2$ | $2$ | $7$ |
| 252.432.11-252.bg.1.3 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.bh.1.1 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.bq.1.3 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.br.1.1 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.eq.1.3 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.er.1.1 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.es.1.3 | $252$ | $2$ | $2$ | $11$ |
| 252.432.11-252.et.1.1 | $252$ | $2$ | $2$ | $11$ |