Modular curve $X_{\mathrm{sp}}(48)$

• Label: 48.4608.337.wu.1
• Level: $48$
• Index: $4608$
• Genus: $337$
• Analytic rank: $89$
• Cusps: $96$
• $\Q$-cusps: $8$

Invariants

Level:	$48$		$\SL_2$-level:	$48$		Newform level:	$2304$
Index:	$4608$		$\PSL_2$-index:	$4608$
Genus:	$337 = 1 + \frac{ 4608 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 96 }{2}$
Cusps:	$96$ (of which $8$ are rational)		Cusp widths	$48^{96}$		Cusp orbits	$1^{8}\cdot2^{8}\cdot4^{6}\cdot8^{4}\cdot16$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$89$
$\Q$-gonality:	$62 \le \gamma \le 64$
$\overline{\Q}$-gonality:	$62 \le \gamma \le 64$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label:

48.4608.337.299

Level structure

$\GL_2(\Z/48\Z)$-generators:	$\begin{bmatrix}1&0\\0&7\end{bmatrix}$, $\begin{bmatrix}1&0\\0&17\end{bmatrix}$, $\begin{bmatrix}17&0\\0&17\end{bmatrix}$, $\begin{bmatrix}19&0\\0&43\end{bmatrix}$, $\begin{bmatrix}31&0\\0&31\end{bmatrix}$, $\begin{bmatrix}31&0\\0&37\end{bmatrix}$
$\GL_2(\Z/48\Z)$-subgroup:	$C_2^4\times C_4^2$
Contains $-I$:	yes
Quadratic refinements:	48.9216.337-48.wu.1.1, 48.9216.337-48.wu.1.2, 48.9216.337-48.wu.1.3, 48.9216.337-48.wu.1.4, 48.9216.337-48.wu.1.5, 48.9216.337-48.wu.1.6, 48.9216.337-48.wu.1.7, 48.9216.337-48.wu.1.8, 48.9216.337-48.wu.1.9, 48.9216.337-48.wu.1.10, 48.9216.337-48.wu.1.11, 48.9216.337-48.wu.1.12, 48.9216.337-48.wu.1.13, 48.9216.337-48.wu.1.14
Cyclic 48-isogeny field degree:	$1$
Cyclic 48-torsion field degree:	$16$
Full 48-torsion field degree:	$256$

Jacobian

Conductor:	$2^{2124}\cdot3^{422}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{293}\cdot2^{20}\cdot4$
Newforms:	24.2.a.a$^{12}$, 32.2.a.a$^{12}$, 36.2.a.a$^{7}$, 48.2.a.a$^{10}$, 64.2.a.a$^{9}$, 72.2.a.a$^{6}$, 96.2.a.a$^{8}$, 96.2.a.b$^{8}$, 128.2.a.a$^{6}$, 128.2.a.b$^{6}$, 128.2.a.c$^{6}$, 128.2.a.d$^{6}$, 144.2.a.a$^{5}$, 144.2.a.b$^{5}$, 192.2.a.a$^{6}$, 192.2.a.b$^{6}$, 192.2.a.c$^{6}$, 192.2.a.d$^{6}$, 256.2.a.a$^{3}$, 256.2.a.b$^{3}$, 256.2.a.c$^{3}$, 256.2.a.d$^{3}$, 256.2.a.e$^{3}$, 288.2.a.a$^{4}$, 288.2.a.b$^{4}$, 288.2.a.c$^{4}$, 288.2.a.d$^{4}$, 288.2.a.e$^{4}$, 384.2.a.a$^{4}$, 384.2.a.b$^{4}$, 384.2.a.c$^{4}$, 384.2.a.d$^{4}$, 384.2.a.e$^{4}$, 384.2.a.f$^{4}$, 384.2.a.g$^{4}$, 384.2.a.h$^{4}$, 576.2.a.a$^{3}$, 576.2.a.b$^{3}$, 576.2.a.c$^{3}$, 576.2.a.d$^{3}$, 576.2.a.e$^{3}$, 576.2.a.f$^{3}$, 576.2.a.g$^{3}$, 576.2.a.h$^{3}$, 576.2.a.i$^{3}$, 768.2.a.a$^{2}$, 768.2.a.b$^{2}$, 768.2.a.c$^{2}$, 768.2.a.d$^{2}$, 768.2.a.e$^{2}$, 768.2.a.f$^{2}$, 768.2.a.g$^{2}$, 768.2.a.h$^{2}$, 768.2.a.i$^{2}$, 768.2.a.j$^{2}$, 768.2.a.k$^{2}$, 768.2.a.l$^{2}$, 1152.2.a.a$^{2}$, 1152.2.a.b$^{2}$, 1152.2.a.c$^{2}$, 1152.2.a.d$^{2}$, 1152.2.a.e$^{2}$, 1152.2.a.f$^{2}$, 1152.2.a.g$^{2}$, 1152.2.a.h$^{2}$, 1152.2.a.i$^{2}$, 1152.2.a.j$^{2}$, 1152.2.a.k$^{2}$, 1152.2.a.l$^{2}$, 1152.2.a.m$^{2}$, 1152.2.a.n$^{2}$, 1152.2.a.o$^{2}$, 1152.2.a.p$^{2}$, 1152.2.a.q$^{2}$, 1152.2.a.r$^{2}$, 1152.2.a.s$^{2}$, 1152.2.a.t$^{2}$, 2304.2.a.a, 2304.2.a.b, 2304.2.a.c, 2304.2.a.d, 2304.2.a.e, 2304.2.a.f, 2304.2.a.g, 2304.2.a.h, 2304.2.a.i, 2304.2.a.j, 2304.2.a.k, 2304.2.a.l, 2304.2.a.m, 2304.2.a.n, 2304.2.a.o, 2304.2.a.p, 2304.2.a.q, 2304.2.a.r, 2304.2.a.s, 2304.2.a.t, 2304.2.a.u, 2304.2.a.v, 2304.2.a.w, 2304.2.a.x, 2304.2.a.y, 2304.2.a.z

Rational points

This modular curve has 8 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
$X_{\mathrm{sp}}(3)$	$3$	$384$	$384$	$0$	$0$	full Jacobian
$X_{\mathrm{sp}}(16)$	$16$	$12$	$12$	$21$	$4$	$1^{274}\cdot2^{19}\cdot4$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.1536.105.do.2	$48$	$3$	$3$	$105$	$21$	$1^{200}\cdot2^{14}\cdot4$
48.2304.161.cav.1	$48$	$2$	$2$	$161$	$43$	$1^{176}$
48.2304.161.cax.1	$48$	$2$	$2$	$161$	$32$	$1^{132}\cdot2^{20}\cdot4$
$X_{\mathrm{sp}}^+(48)$	$48$	$2$	$2$	$161$	$55$	$1^{150}\cdot2^{11}\cdot4$
48.2304.169.bii.2	$48$	$2$	$2$	$169$	$42$	$1^{148}\cdot2^{10}$
48.2304.169.bit.2	$48$	$2$	$2$	$169$	$44$	$1^{148}\cdot2^{10}$
48.2304.169.biw.4	$48$	$2$	$2$	$169$	$45$	$1^{144}\cdot2^{10}\cdot4$
48.2304.169.egl.1	$48$	$2$	$2$	$169$	$48$	$1^{146}\cdot2^{9}\cdot4$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
48.9216.673.sw.1	$48$	$2$	$2$	$673$	$89$	$2^{20}\cdot4^{68}\cdot8^{3}$
48.9216.673.baq.1	$48$	$2$	$2$	$673$	$99$	$2^{128}\cdot4^{20}$
48.9216.673.bba.1	$48$	$2$	$2$	$673$	$101$	$2^{128}\cdot4^{20}$
48.9216.673.bbh.1	$48$	$2$	$2$	$673$	$95$	$2^{54}\cdot4^{53}\cdot8^{2}$
48.9216.673.bbr.1	$48$	$2$	$2$	$673$	$95$	$2^{54}\cdot4^{53}\cdot8^{2}$
48.9216.673.bcb.1	$48$	$2$	$2$	$673$	$101$	$2^{128}\cdot4^{20}$
48.9216.673.bcn.1	$48$	$2$	$2$	$673$	$99$	$2^{128}\cdot4^{20}$
48.9216.673.beg.1	$48$	$2$	$2$	$673$	$89$	$2^{20}\cdot4^{68}\cdot8^{3}$