Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AN9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.9.8906 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&46\\24&25\end{bmatrix}$, $\begin{bmatrix}17&32\\24&5\end{bmatrix}$, $\begin{bmatrix}25&29\\24&19\end{bmatrix}$, $\begin{bmatrix}29&15\\0&31\end{bmatrix}$, $\begin{bmatrix}31&9\\12&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.bhx.4 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{47}\cdot3^{9}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}\cdot4$ |
| Newforms: | 24.2.a.a, 96.2.d.a, 96.2.f.a, 192.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v + y z $ |
| $=$ | $y u - y s - t^{2} + t u + u r + r^{2}$ | |
| $=$ | $y^{2} + y v - 2 y s + t v - u v - v r$ | |
| $=$ | $y u - y s + t^{2} + t u + u r - v^{2} - 2 v r - r^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{8} y^{2} - x^{6} y^{4} - 12 x^{6} z^{4} + x^{4} y^{6} - 18 x^{4} y^{4} z^{2} + 28 x^{4} y^{2} z^{4} + \cdots + 8 y^{6} z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.gf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-XY^{3}-X^{2}Z^{2}-Y^{2}Z^{2}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.9.bhx.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{8}Y^{2}-X^{6}Y^{4}-12X^{6}Z^{4}+X^{4}Y^{6}-18X^{4}Y^{4}Z^{2}+28X^{4}Y^{2}Z^{4}-24X^{4}Z^{6}+6X^{2}Y^{6}Z^{2}-32X^{2}Y^{4}Z^{4}+8X^{2}Y^{2}Z^{6}+8Y^{6}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-24.gf.2.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.qe.4.15 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.qe.4.38 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.5-48.ot.2.34 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.192.5-48.ot.2.49 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.17-48.gx.1.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.kh.5.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.rd.1.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.ri.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bcq.2.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bcv.2.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdf.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdm.4.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.beb.5.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bej.5.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bfl.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bfn.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bjm.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bjn.2.15 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bkb.4.11 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bke.4.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bpa.4.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bpb.4.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bpp.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bps.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brk.4.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brp.4.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brz.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bsg.2.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.1152.33-48.chz.2.15 | $48$ | $3$ | $3$ | $33$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{3}$ |