Invariants
Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
Genus: | $13 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
Cusps: | $40$ (none of which are rational) | Cusp widths | $1^{8}\cdot2^{4}\cdot3^{8}\cdot4^{4}\cdot6^{4}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $2^{8}\cdot4^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-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: | 48AH13 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.13.10587 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&29\\24&41\end{bmatrix}$, $\begin{bmatrix}19&1\\12&23\end{bmatrix}$, $\begin{bmatrix}23&27\\36&47\end{bmatrix}$, $\begin{bmatrix}37&18\\36&11\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.384.13.ok.2 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $2$ |
Cyclic 48-torsion field degree: | $4$ |
Full 48-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{51}\cdot3^{13}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{3}\cdot2^{5}$ |
Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 24.2.f.a, 48.2.c.a, 96.2.f.a, 192.2.a.b, 192.2.a.d |
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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.384.5-24.gg.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
48.384.5-24.gg.2.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
48.384.5-48.ka.3.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
48.384.5-48.ka.3.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
48.384.5-48.la.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
48.384.5-48.la.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
48.384.7-48.es.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
48.384.7-48.es.1.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
48.384.7-48.ey.3.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
48.384.7-48.ey.3.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
48.384.7-48.hm.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
48.384.7-48.hm.1.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
48.384.7-48.hw.1.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
48.384.7-48.hw.1.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
48.1536.33-48.fs.4.2 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.hv.2.12 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.is.2.12 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.kv.2.12 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.uc.4.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.up.2.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.vu.2.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
48.1536.33-48.wb.2.6 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
48.2304.57-48.ie.2.4 | $48$ | $3$ | $3$ | $57$ | $2$ | $1^{10}\cdot2^{13}\cdot4^{2}$ |