Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12\cdot48$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.5.383 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&5\\16&9\end{bmatrix}$, $\begin{bmatrix}5&8\\20&15\end{bmatrix}$, $\begin{bmatrix}25&43\\16&15\end{bmatrix}$, $\begin{bmatrix}35&38\\32&27\end{bmatrix}$, $\begin{bmatrix}41&33\\36&5\end{bmatrix}$, $\begin{bmatrix}47&16\\24&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.5.f.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $8192$ |
Jacobian
| Conductor: | $2^{19}\cdot3^{10}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 36.2.a.a$^{2}$, 288.2.a.a, 288.2.a.d, 288.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ x w v + t u v $ |
| $=$ | $x w v + w^{2} u$ | |
| $=$ | $x w v - y v^{2}$ | |
| $=$ | $ - x^{2} v + t u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{11} - 2 x y^{2} z^{8} - 27 y z^{10} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ 11664 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:1:0:0:0:0)$, $(0:-1:1:0:0:0:0)$, $(0:1:1:0:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^2}\cdot\frac{108yz^{4}tv+39yzuv^{4}+76yt^{3}v^{3}+4yv^{6}-27z^{7}-144z^{3}t^{2}v^{2}}{vtz^{4}y}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.72.5.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 18z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}v$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{11}-2XY^{2}Z^{8}-27YZ^{10} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.72.5.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 12zuv^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -u$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.72.2-24.cj.1.30 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 48.48.1-48.b.1.11 | $48$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 48.72.2-24.cj.1.30 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.9-48.g.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.v.1.5 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.be.1.33 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.bm.1.9 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.hm.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.hn.1.25 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.hq.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.hr.1.9 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.ic.1.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.id.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.ig.1.18 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.ih.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.jg.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jg.2.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jh.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jh.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ji.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ji.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jj.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jj.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jk.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jk.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jl.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jl.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jm.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jm.2.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jn.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jn.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jo.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jo.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jp.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jp.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jq.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jq.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jr.1.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jr.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.js.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.js.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jt.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jt.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ju.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ju.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jv.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.jv.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ko.1.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.kp.1.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.ks.1.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.kt.1.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.le.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.lf.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.li.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.lj.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.ls.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.lv.1.9 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.lw.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.lz.1.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 144.432.17-144.f.1.1 | $144$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 240.288.9-240.is.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.it.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.iw.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ix.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jy.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jz.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.kc.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.kd.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ko.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.kp.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ks.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.kt.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ls.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ls.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lt.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lt.2.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lu.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lu.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lv.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lv.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lw.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lw.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lx.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lx.2.20 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ly.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ly.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lz.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.lz.2.20 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ma.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ma.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mb.1.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mb.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mc.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mc.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.md.1.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.md.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.me.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.me.2.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mf.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mf.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mg.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mg.2.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mh.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mh.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ne.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.nf.1.20 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ni.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.nj.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.nu.1.44 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.nv.1.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ny.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.nz.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.pa.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.pb.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.pe.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.pf.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |