Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| 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: | 48.384.9.8902 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&45\\24&35\end{bmatrix}$, $\begin{bmatrix}7&26\\36&41\end{bmatrix}$, $\begin{bmatrix}11&4\\12&41\end{bmatrix}$, $\begin{bmatrix}31&14\\12&29\end{bmatrix}$, $\begin{bmatrix}31&46\\36&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.bib.1 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: | $2^{43}\cdot3^{9}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}\cdot4$ |
| Newforms: | 24.2.a.a, 96.2.c.a, 96.2.d.a, 96.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v - y w $ |
| $=$ | $z s - u s + v r$ | |
| $=$ | $z v - w r + u v + r s$ | |
| $=$ | $x v - y s - z v - t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 18 x^{12} + 12 x^{10} y^{2} - 27 x^{10} z^{2} + 16 x^{8} y^{4} + 54 x^{8} y^{2} z^{2} + \cdots + 2 y^{6} z^{6} $ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -y-r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
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.bib.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -18X^{12}+12X^{10}Y^{2}-27X^{10}Z^{2}+16X^{8}Y^{4}+54X^{8}Y^{2}Z^{2}-9X^{8}Z^{4}+24X^{6}Y^{2}Z^{4}-14X^{4}Y^{4}Z^{4}+4X^{2}Y^{6}Z^{4}-X^{2}Y^{4}Z^{6}+2Y^{6}Z^{6} $ |
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.2.22 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.qe.1.8 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.qe.1.36 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.5-48.ou.1.18 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.192.5-48.ou.1.61 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
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.gp.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.kv.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.rb.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.rm.2.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bcy.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdd.3.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdn.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdu.1.15 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdz.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bep.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bfj.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bfp.1.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bju.3.9 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bjv.3.10 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bkj.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bkm.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bpi.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bpj.1.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bpx.3.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bqa.3.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brs.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brx.1.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bsh.3.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bso.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.1152.33-48.chv.2.15 | $48$ | $3$ | $3$ | $33$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{3}$ |