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Properties

Label 48.4608.161-48.kxx.2.1
Level $48$
Index $4608$
Genus $161$
Analytic rank $29$
Cusps $64$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $2304$
Index: $4608$ $\PSL_2$-index:$2304$
Genus: $161 = 1 + \frac{ 2304 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 64 }{2}$
Cusps: $64$ (none of which are rational) Cusp widths $24^{32}\cdot48^{32}$ Cusp orbits $2^{2}\cdot4^{13}\cdot8$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $29$
$\Q$-gonality: $24 \le \gamma \le 32$
$\overline{\Q}$-gonality: $24 \le \gamma \le 32$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.4608.161.425

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&0\\12&13\end{bmatrix}$, $\begin{bmatrix}11&45\\0&25\end{bmatrix}$, $\begin{bmatrix}31&3\\12&25\end{bmatrix}$
$\GL_2(\Z/48\Z)$-subgroup: $\OD_{16}:\OD_{16}$
Contains $-I$: no $\quad$ (see 48.2304.161.kxx.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $4$
Cyclic 48-torsion field degree: $16$
Full 48-torsion field degree: $256$

Jacobian

Conductor: $2^{1089}\cdot3^{230}$
Simple: no
Squarefree: no
Decomposition: $1^{57}\cdot2^{40}\cdot4^{6}$
Newforms: 24.2.a.a, 36.2.a.a$^{2}$, 36.2.b.a$^{3}$, 48.2.a.a$^{3}$, 48.2.c.a$^{2}$, 64.2.a.a$^{3}$, 72.2.a.a$^{2}$, 144.2.a.a$^{2}$, 144.2.a.b$^{3}$, 144.2.c.a, 192.2.a.a$^{2}$, 192.2.a.c$^{2}$, 192.2.a.d$^{2}$, 192.2.c.a$^{2}$, 192.2.c.b$^{2}$, 256.2.a.b, 256.2.a.c, 256.2.a.e$^{3}$, 576.2.a.a, 576.2.a.c, 576.2.a.d$^{4}$, 576.2.a.e$^{2}$, 576.2.a.g, 576.2.a.h, 576.2.a.i, 576.2.c.a, 576.2.c.b, 576.2.c.c, 768.2.a.b, 768.2.a.c, 768.2.a.f, 768.2.a.g, 768.2.a.i$^{2}$, 768.2.a.l$^{2}$, 768.2.c.a$^{2}$, 768.2.c.b$^{2}$, 768.2.c.c$^{2}$, 768.2.c.d$^{2}$, 768.2.c.e$^{2}$, 768.2.c.f$^{2}$, 768.2.c.j$^{2}$, 2304.2.a.a$^{3}$, 2304.2.a.c, 2304.2.a.d, 2304.2.a.f$^{2}$, 2304.2.a.g$^{2}$, 2304.2.a.j$^{2}$, 2304.2.a.k$^{2}$, 2304.2.a.m, 2304.2.a.n, 2304.2.a.p$^{3}$, 2304.2.a.r, 2304.2.a.t, 2304.2.a.x, 2304.2.a.z, 2304.2.c.a, 2304.2.c.b, 2304.2.c.c, 2304.2.c.d, 2304.2.c.e, 2304.2.c.f, 2304.2.c.g, 2304.2.c.h

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=17,19,\ldots,401$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.2304.65-24.bcd.1.1 $24$ $2$ $2$ $65$ $5$ $1^{24}\cdot2^{30}\cdot4^{3}$
48.2304.65-24.bcd.1.4 $48$ $2$ $2$ $65$ $5$ $1^{24}\cdot2^{30}\cdot4^{3}$
48.1536.49-48.ckg.1.8 $48$ $3$ $3$ $49$ $9$ $1^{40}\cdot2^{28}\cdot4^{4}$
48.1536.49-48.ckg.3.8 $48$ $3$ $3$ $49$ $9$ $1^{40}\cdot2^{28}\cdot4^{4}$
48.2304.81-48.mhx.2.1 $48$ $2$ $2$ $81$ $23$ $2^{30}\cdot4^{5}$
48.2304.81-48.mhx.2.2 $48$ $2$ $2$ $81$ $23$ $2^{30}\cdot4^{5}$
48.2304.81-48.mhy.2.1 $48$ $2$ $2$ $81$ $11$ $1^{24}\cdot2^{20}\cdot4^{4}$
48.2304.81-48.mhy.2.9 $48$ $2$ $2$ $81$ $11$ $1^{24}\cdot2^{20}\cdot4^{4}$
48.2304.81-48.moj.1.4 $48$ $2$ $2$ $81$ $11$ $1^{40}\cdot2^{14}\cdot4^{3}$
48.2304.81-48.moj.1.10 $48$ $2$ $2$ $81$ $11$ $1^{40}\cdot2^{14}\cdot4^{3}$
48.2304.81-48.mol.2.7 $48$ $2$ $2$ $81$ $19$ $1^{16}\cdot2^{20}\cdot4^{6}$
48.2304.81-48.mol.2.11 $48$ $2$ $2$ $81$ $19$ $1^{16}\cdot2^{20}\cdot4^{6}$
48.2304.81-48.mos.1.1 $48$ $2$ $2$ $81$ $13$ $1^{40}\cdot2^{16}\cdot4^{2}$
48.2304.81-48.mos.1.13 $48$ $2$ $2$ $81$ $13$ $1^{40}\cdot2^{16}\cdot4^{2}$
48.2304.81-48.mou.2.1 $48$ $2$ $2$ $81$ $17$ $1^{16}\cdot2^{30}\cdot4$
48.2304.81-48.mou.2.13 $48$ $2$ $2$ $81$ $17$ $1^{16}\cdot2^{30}\cdot4$