Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot6^{2}\cdot12\cdot16\cdot48$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48C5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.5.4005 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&43\\36&11\end{bmatrix}$, $\begin{bmatrix}11&18\\12&17\end{bmatrix}$, $\begin{bmatrix}37&14\\0&1\end{bmatrix}$, $\begin{bmatrix}47&3\\12&47\end{bmatrix}$, $\begin{bmatrix}47&38\\24&19\end{bmatrix}$, $\begin{bmatrix}47&46\\36&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.ou.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{23}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot4$ |
| Newforms: | 24.2.a.a, 96.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x t + y w - y t $ |
| $=$ | $x y + 2 y^{2} - w^{2} - w t$ | |
| $=$ | $ - x w - 2 x t + y w - y t + 3 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{5} z^{2} + 3 x^{4} y^{2} z - 2 x^{3} y^{4} + 4 x^{3} z^{4} - 3 x y^{4} z^{2} + y^{6} z $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1:0:0:0:0)$, $(-2:1:0:0:0)$, $(0:-1:0:1:1)$, $(0:0:0:0:1)$, $(0:0:0:-1:1)$, $(0:1:0:1:1)$, $(4:1:0:-3:1)$, $(-4:-1:0:-3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}-6x^{10}w^{2}-3x^{10}wt+112x^{10}t^{2}-374x^{8}w^{2}t^{2}-1044x^{8}wt^{3}-1774x^{8}t^{4}+4200x^{6}w^{2}t^{4}+22947x^{6}wt^{5}+103428x^{6}t^{6}-434538x^{4}w^{2}t^{6}-2584512x^{4}wt^{7}-12092550x^{4}t^{8}+52037620x^{2}w^{2}t^{8}+306432520x^{2}wt^{9}+1251263924x^{2}t^{10}-4095y^{12}+243986y^{10}t^{2}-6943169y^{8}t^{4}+130013084y^{6}t^{6}-1608257985y^{4}t^{8}+6957372562y^{2}t^{10}-2736212160w^{2}t^{10}-2736212160wt^{11}+t^{12}}{t^{2}(x^{4}t^{6}-13x^{2}w^{2}t^{6}-107x^{2}wt^{7}-504x^{2}t^{8}-y^{10}+10y^{8}t^{2}-81y^{6}t^{4}+704y^{4}t^{6}-2904y^{2}t^{8}+1136w^{2}t^{8}+1136wt^{9})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.ou.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{5}Z^{2}+3X^{4}Y^{2}Z-2X^{3}Y^{4}+4X^{3}Z^{4}-3XY^{4}Z^{2}+Y^{6}Z $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.ir.1.43 | $24$ | $2$ | $2$ | $1$ | $0$ | $4$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.9-48.hp.4.39 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.iz.1.19 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.mi.2.45 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.mp.1.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bbm.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bbm.4.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bbo.2.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 48.384.9-48.bbo.4.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 48.384.9-48.bef.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bei.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bej.2.25 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bem.2.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bgc.2.21 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bgc.4.13 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bge.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bge.4.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bhd.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhd.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhe.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhe.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhf.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhf.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhg.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhg.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhp.2.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhp.4.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhq.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhq.4.13 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhr.1.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhr.3.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhs.1.6 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhs.3.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bht.1.6 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bht.3.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhu.1.6 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhu.3.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhv.2.6 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhv.4.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhw.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhw.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bib.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bib.4.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bic.1.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bic.3.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bid.1.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bid.3.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bie.2.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bie.4.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.big.2.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.big.4.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bii.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bii.4.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bio.2.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bio.4.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.biq.2.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.biq.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.576.17-48.caa.1.33 | $48$ | $3$ | $3$ | $17$ | $0$ | $1^{4}\cdot4^{2}$ |
| 240.384.9-240.fxz.2.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fya.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyd.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fye.1.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyp.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyp.4.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyq.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyq.4.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzf.2.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzg.2.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzj.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzk.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzv.2.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzv.4.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzw.2.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzw.4.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfd.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfd.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfe.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfe.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gff.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gff.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfg.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfg.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfp.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfp.4.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfq.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfq.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfr.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfr.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfs.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfs.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gft.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gft.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfu.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfu.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfv.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfv.4.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfw.3.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfw.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggb.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggb.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggc.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggc.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggd.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggd.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gge.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gge.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggp.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggp.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggq.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggq.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghf.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghf.4.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghg.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghg.4.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |