Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| 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: | 48.192.5.4413 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&6\\0&29\end{bmatrix}$, $\begin{bmatrix}5&45\\36&41\end{bmatrix}$, $\begin{bmatrix}11&27\\0&25\end{bmatrix}$, $\begin{bmatrix}17&4\\24&29\end{bmatrix}$, $\begin{bmatrix}29&31\\12&17\end{bmatrix}$, $\begin{bmatrix}47&16\\0&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.ot.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: | $2^{27}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot4$ |
| Newforms: | 24.2.a.a, 192.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x t + z w + z t $ |
| $=$ | $x z - 2 z^{2} + w^{2} - w t$ | |
| $=$ | $ - x w + 2 x t + 6 y^{2} - z w - z t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{5} z^{2} + 12 x^{4} y^{2} z - 8 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 |
|---|
| $(1:0:0:0:0)$, $(2:0:1:0:0)$, $(0:0:-1:-1:1)$, $(0:0:0:0:1)$, $(0:0:0:1:1)$, $(0:0:1:-1:1)$, $(-4:0:1:3:1)$, $(4:0:-1:3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}-6x^{10}w^{2}+3x^{10}wt+112x^{10}t^{2}-374x^{8}w^{2}t^{2}+1044x^{8}wt^{3}-1774x^{8}t^{4}+4200x^{6}w^{2}t^{4}-22947x^{6}wt^{5}+103428x^{6}t^{6}-434538x^{4}w^{2}t^{6}+2584512x^{4}wt^{7}-12092550x^{4}t^{8}+52037620x^{2}w^{2}t^{8}-306432520x^{2}wt^{9}+1251263924x^{2}t^{10}-4095z^{12}+243986z^{10}t^{2}-6943169z^{8}t^{4}+130013084z^{6}t^{6}-1608257985z^{4}t^{8}+6957372562z^{2}t^{10}-2736212160w^{2}t^{10}+2736212160wt^{11}+t^{12}}{t^{2}(x^{4}t^{6}-13x^{2}w^{2}t^{6}+107x^{2}wt^{7}-504x^{2}t^{8}-z^{10}+10z^{8}t^{2}-81z^{6}t^{4}+704z^{4}t^{6}-2904z^{2}t^{8}+1136w^{2}t^{8}-1136wt^{9})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.ot.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{4}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -4X^{5}Z^{2}+12X^{4}Y^{2}Z-8X^{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.1.44 | $24$ | $2$ | $2$ | $1$ | $0$ | $4$ |
| 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.hq.4.32 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.ix.2.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.mj.2.23 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.mo.2.22 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bbl.2.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bbl.4.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bbn.2.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 48.384.9-48.bbn.4.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 48.384.9-48.beg.2.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.beh.2.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bek.2.13 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bel.2.26 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bgb.2.25 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bgb.4.21 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bgd.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bgd.4.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bgz.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bgz.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bha.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bha.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhb.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhb.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhc.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhc.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhh.2.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhh.4.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhi.3.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhi.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhj.1.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhj.3.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhk.1.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhk.3.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhl.1.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhl.3.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhm.1.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhm.3.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhn.2.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhn.4.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bho.3.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bho.4.13 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhx.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhx.4.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhy.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhy.3.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhz.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bhz.3.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bia.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bia.4.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.384.9-48.bif.2.19 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bif.4.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bih.2.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bih.4.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.bin.2.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bin.4.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bip.2.25 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bip.4.19 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 48.576.17-48.cab.1.33 | $48$ | $3$ | $3$ | $17$ | $0$ | $1^{4}\cdot4^{2}$ |
| 240.384.9-240.fxx.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fxy.2.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyb.2.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyc.2.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyn.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyn.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyo.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fyo.4.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzd.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fze.2.18 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzh.2.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzi.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzt.2.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzt.4.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzu.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fzu.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gez.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gez.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfa.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfa.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfb.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfb.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfc.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfc.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfh.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfh.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfi.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfi.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfj.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfj.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfk.2.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfk.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfl.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfl.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfm.2.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfm.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfn.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfn.4.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfo.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfo.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfx.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfx.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfy.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfy.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfz.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gfz.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gga.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gga.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggn.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggn.4.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggo.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ggo.4.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghd.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghd.4.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghe.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ghe.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |