Modular curve 42.252.16.a.1

• Label: 42.252.16.a.1
• Level: $42$
• Index: $252$
• Genus: $16$
• Analytic rank: $4$
• Cusps: $12$
• $\Q$-cusps: $0$

Invariants

Level:	$42$		$\SL_2$-level:	$42$		Newform level:	$294$
Index:	$252$		$\PSL_2$-index:	$252$
Genus:	$16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (none of which are rational)		Cusp widths	$7^{3}\cdot14^{3}\cdot21^{3}\cdot42^{3}$		Cusp orbits	$3^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$4$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	42A16
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	42.252.16.1

Level structure

$\GL_2(\Z/42\Z)$-generators:	$\begin{bmatrix}5&26\\12&35\end{bmatrix}$, $\begin{bmatrix}5&41\\6&41\end{bmatrix}$, $\begin{bmatrix}25&11\\6&31\end{bmatrix}$, $\begin{bmatrix}31&1\\36&23\end{bmatrix}$, $\begin{bmatrix}31&5\\6&11\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	42.504.16-42.a.1.1, 42.504.16-42.a.1.2, 42.504.16-42.a.1.3, 42.504.16-42.a.1.4, 42.504.16-42.a.1.5, 42.504.16-42.a.1.6, 42.504.16-42.a.1.7, 42.504.16-42.a.1.8, 42.504.16-42.a.1.9, 42.504.16-42.a.1.10, 42.504.16-42.a.1.11, 42.504.16-42.a.1.12, 42.504.16-42.a.1.13, 42.504.16-42.a.1.14, 42.504.16-42.a.1.15, 42.504.16-42.a.1.16, 84.504.16-42.a.1.1, 84.504.16-42.a.1.2, 84.504.16-42.a.1.3, 84.504.16-42.a.1.4, 84.504.16-42.a.1.5, 84.504.16-42.a.1.6, 84.504.16-42.a.1.7, 84.504.16-42.a.1.8, 84.504.16-42.a.1.9, 84.504.16-42.a.1.10, 84.504.16-42.a.1.11, 84.504.16-42.a.1.12, 84.504.16-42.a.1.13, 84.504.16-42.a.1.14, 84.504.16-42.a.1.15, 84.504.16-42.a.1.16, 84.504.16-42.a.1.17, 84.504.16-42.a.1.18, 84.504.16-42.a.1.19, 84.504.16-42.a.1.20, 84.504.16-42.a.1.21, 84.504.16-42.a.1.22, 84.504.16-42.a.1.23, 84.504.16-42.a.1.24, 84.504.16-42.a.1.25, 84.504.16-42.a.1.26, 84.504.16-42.a.1.27, 84.504.16-42.a.1.28, 84.504.16-42.a.1.29, 84.504.16-42.a.1.30, 84.504.16-42.a.1.31, 84.504.16-42.a.1.32, 84.504.16-42.a.1.33, 84.504.16-42.a.1.34, 84.504.16-42.a.1.35, 84.504.16-42.a.1.36, 84.504.16-42.a.1.37, 84.504.16-42.a.1.38, 84.504.16-42.a.1.39, 84.504.16-42.a.1.40, 84.504.16-42.a.1.41, 84.504.16-42.a.1.42, 84.504.16-42.a.1.43, 84.504.16-42.a.1.44, 84.504.16-42.a.1.45, 84.504.16-42.a.1.46, 84.504.16-42.a.1.47, 84.504.16-42.a.1.48, 168.504.16-42.a.1.1, 168.504.16-42.a.1.2, 168.504.16-42.a.1.3, 168.504.16-42.a.1.4, 168.504.16-42.a.1.5, 168.504.16-42.a.1.6, 168.504.16-42.a.1.7, 168.504.16-42.a.1.8, 168.504.16-42.a.1.9, 168.504.16-42.a.1.10, 168.504.16-42.a.1.11, 168.504.16-42.a.1.12, 168.504.16-42.a.1.13, 168.504.16-42.a.1.14, 168.504.16-42.a.1.15, 168.504.16-42.a.1.16, 168.504.16-42.a.1.17, 168.504.16-42.a.1.18, 168.504.16-42.a.1.19, 168.504.16-42.a.1.20, 168.504.16-42.a.1.21, 168.504.16-42.a.1.22, 168.504.16-42.a.1.23, 168.504.16-42.a.1.24, 168.504.16-42.a.1.25, 168.504.16-42.a.1.26, 168.504.16-42.a.1.27, 168.504.16-42.a.1.28, 168.504.16-42.a.1.29, 168.504.16-42.a.1.30, 168.504.16-42.a.1.31, 168.504.16-42.a.1.32, 168.504.16-42.a.1.33, 168.504.16-42.a.1.34, 168.504.16-42.a.1.35, 168.504.16-42.a.1.36, 168.504.16-42.a.1.37, 168.504.16-42.a.1.38, 168.504.16-42.a.1.39, 168.504.16-42.a.1.40, 168.504.16-42.a.1.41, 168.504.16-42.a.1.42, 168.504.16-42.a.1.43, 168.504.16-42.a.1.44, 168.504.16-42.a.1.45, 168.504.16-42.a.1.46, 168.504.16-42.a.1.47, 168.504.16-42.a.1.48, 168.504.16-42.a.1.49, 168.504.16-42.a.1.50, 168.504.16-42.a.1.51, 168.504.16-42.a.1.52, 168.504.16-42.a.1.53, 168.504.16-42.a.1.54, 168.504.16-42.a.1.55, 168.504.16-42.a.1.56, 168.504.16-42.a.1.57, 168.504.16-42.a.1.58, 168.504.16-42.a.1.59, 168.504.16-42.a.1.60, 168.504.16-42.a.1.61, 168.504.16-42.a.1.62, 168.504.16-42.a.1.63, 168.504.16-42.a.1.64, 210.504.16-42.a.1.1, 210.504.16-42.a.1.2, 210.504.16-42.a.1.3, 210.504.16-42.a.1.4, 210.504.16-42.a.1.5, 210.504.16-42.a.1.6, 210.504.16-42.a.1.7, 210.504.16-42.a.1.8, 210.504.16-42.a.1.9, 210.504.16-42.a.1.10, 210.504.16-42.a.1.11, 210.504.16-42.a.1.12, 210.504.16-42.a.1.13, 210.504.16-42.a.1.14, 210.504.16-42.a.1.15, 210.504.16-42.a.1.16
Cyclic 42-isogeny field degree:	$8$
Cyclic 42-torsion field degree:	$96$
Full 42-torsion field degree:	$2304$

Jacobian

Conductor:	$2^{6}\cdot3^{12}\cdot7^{32}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{4}\cdot2^{6}$
Newforms:	98.2.a.b$^{2}$, 147.2.a.c$^{2}$, 147.2.a.d$^{2}$, 147.2.a.e$^{2}$, 294.2.a.d, 294.2.a.e

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	Kernel decomposition
$X_0(2)$	$2$	$84$	$84$	$0$	$0$	full Jacobian
$X_0(3)$	$3$	$63$	$63$	$0$	$0$	full Jacobian
$X_{\mathrm{ns}}^+(7)$	$7$	$12$	$12$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_0(6)$	$6$	$21$	$21$	$0$	$0$	full Jacobian
14.63.2.a.1	$14$	$4$	$4$	$2$	$0$	$1^{4}\cdot2^{5}$
21.84.5.a.1	$21$	$3$	$3$	$5$	$2$	$1^{3}\cdot2^{4}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
42.504.31.a.1	$42$	$2$	$2$	$31$	$5$	$1^{15}$
42.504.31.b.1	$42$	$2$	$2$	$31$	$14$	$1^{15}$
42.504.31.c.1	$42$	$2$	$2$	$31$	$5$	$1^{15}$
42.504.31.d.1	$42$	$2$	$2$	$31$	$9$	$1^{15}$
42.504.34.a.1	$42$	$2$	$2$	$34$	$6$	$1^{6}\cdot2^{6}$
42.504.34.c.1	$42$	$2$	$2$	$34$	$12$	$1^{6}\cdot2^{6}$
42.504.34.d.1	$42$	$2$	$2$	$34$	$8$	$1^{6}\cdot2^{6}$
42.504.34.e.1	$42$	$2$	$2$	$34$	$10$	$1^{6}\cdot2^{6}$
42.504.34.f.1	$42$	$2$	$2$	$34$	$11$	$1^{18}$
42.504.34.g.1	$42$	$2$	$2$	$34$	$6$	$1^{18}$
42.504.34.h.1	$42$	$2$	$2$	$34$	$10$	$1^{18}$
42.504.34.i.1	$42$	$2$	$2$	$34$	$9$	$1^{18}$
42.756.52.a.1	$42$	$3$	$3$	$52$	$14$	$1^{12}\cdot2^{12}$