Properties

Label 126.216.2-126.a.1.7
Level $126$
Index $216$
Genus $2$
Cusps $16$
$\Q$-cusps $0$

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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: $\begin{bmatrix}41&102\\24&13\end{bmatrix}$, $\begin{bmatrix}97&39\\54&83\end{bmatrix}$, $\begin{bmatrix}121&105\\108&43\end{bmatrix}$
Contains $-I$: no $\quad$ (see 126.108.2.a.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.a.1.4 $63$ $3$ $3$ $0$ $0$

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.a.1.4 $63$ $3$ $3$ $0$ $0$

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.a.1.4 $126$ $2$ $2$ $7$
126.432.7-126.ch.1.4 $126$ $2$ $2$ $7$
126.432.7-126.cj.1.4 $126$ $2$ $2$ $7$
126.432.7-126.cl.1.4 $126$ $2$ $2$ $7$
252.432.7-252.c.1.9 $252$ $2$ $2$ $7$
252.432.7-252.m.1.10 $252$ $2$ $2$ $7$
252.432.7-252.n.1.20 $252$ $2$ $2$ $7$
252.432.7-252.cg.1.5 $252$ $2$ $2$ $7$
252.432.7-252.ch.1.10 $252$ $2$ $2$ $7$
252.432.7-252.ci.1.10 $252$ $2$ $2$ $7$
252.432.7-252.cq.1.12 $252$ $2$ $2$ $7$
252.432.7-252.cr.1.11 $252$ $2$ $2$ $7$
252.432.7-252.cs.1.9 $252$ $2$ $2$ $7$
252.432.7-252.da.1.11 $252$ $2$ $2$ $7$
252.432.7-252.db.1.10 $252$ $2$ $2$ $7$
252.432.7-252.dc.1.9 $252$ $2$ $2$ $7$
252.432.11-252.bi.1.10 $252$ $2$ $2$ $11$
252.432.11-252.bj.1.10 $252$ $2$ $2$ $11$
252.432.11-252.bu.1.12 $252$ $2$ $2$ $11$
252.432.11-252.bv.1.10 $252$ $2$ $2$ $11$
252.432.11-252.cm.1.10 $252$ $2$ $2$ $11$
252.432.11-252.cn.1.11 $252$ $2$ $2$ $11$
252.432.11-252.de.1.11 $252$ $2$ $2$ $11$
252.432.11-252.df.1.20 $252$ $2$ $2$ $11$