Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}\cdot16\cdot48$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.5834 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&38\\0&29\end{bmatrix}$, $\begin{bmatrix}11&1\\36&19\end{bmatrix}$, $\begin{bmatrix}13&45\\0&7\end{bmatrix}$, $\begin{bmatrix}17&1\\0&35\end{bmatrix}$, $\begin{bmatrix}25&3\\36&17\end{bmatrix}$, $\begin{bmatrix}43&24\\24&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.qe.3 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: | $2^{13}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 96.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 2 x^{4} - x^{2} y^{2} - 2 x^{2} y z + 2 x^{2} z^{2} + y^{3} z - y^{2} z^{2} - 2 y z^{3} $ |
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.
| Canonical model |
|---|
| $(0:-1:1)$, $(0:1:0)$, $(0:2:1)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{262143x^{2}y^{22}-2620710x^{2}y^{21}z+9775674x^{2}y^{20}z^{2}+5826816x^{2}y^{19}z^{3}-541456092x^{2}y^{18}z^{4}+7329954168x^{2}y^{17}z^{5}-60587704296x^{2}y^{16}z^{6}+331425551232x^{2}y^{15}z^{7}-1247190228576x^{2}y^{14}z^{8}+3289932552384x^{2}y^{13}z^{9}-6103682376768x^{2}y^{12}z^{10}+7871314475520x^{2}y^{11}z^{11}-6947600363136x^{2}y^{10}z^{12}+4220739258624x^{2}y^{9}z^{13}-2040474025728x^{2}y^{8}z^{14}+1031785924608x^{2}y^{7}z^{15}-498210538752x^{2}y^{6}z^{16}+163627872768x^{2}y^{5}z^{17}-34757254656x^{2}y^{4}z^{18}+3884949504x^{2}y^{3}z^{19}-110840832x^{2}y^{2}z^{20}+1382400x^{2}yz^{21}+129024x^{2}z^{22}-131072y^{24}+1310721y^{23}z-4981467y^{22}z^{2}+5953694y^{21}z^{3}-3389718y^{20}z^{4}+488193972y^{19}z^{5}-7728376316y^{18}z^{6}+66978939384y^{17}z^{7}-379827865800y^{16}z^{8}+1484725809248y^{15}z^{9}-4083846570528y^{14}z^{10}+7949616350016y^{13}z^{11}-10900297298240y^{12}z^{12}+10479225205632y^{11}z^{13}-7255895872128y^{10}z^{14}+4087666490624y^{9}z^{15}-2257586576640y^{8}z^{16}+1154339843328y^{7}z^{17}-473687211776y^{6}z^{18}+157964822016y^{5}z^{19}-35996708352y^{4}z^{20}+4000621568y^{3}z^{21}-89213952y^{2}z^{22}+595968yz^{23}-2048z^{24}}{zy(y-2z)^{3}(y+z)^{3}(x^{2}y^{14}+18x^{2}y^{13}z+146x^{2}y^{12}z^{2}+672x^{2}y^{11}z^{3}+1636x^{2}y^{10}z^{4}-392x^{2}y^{9}z^{5}-21048x^{2}y^{8}z^{6}+168960x^{2}y^{7}z^{7}+55728x^{2}y^{6}z^{8}+2656x^{2}y^{5}z^{9}-46752x^{2}y^{4}z^{10}-9472x^{2}y^{3}z^{11}-2112x^{2}y^{2}z^{12}+2176x^{2}yz^{13}-128x^{2}z^{14}-y^{15}z-17y^{14}z^{2}-126y^{13}z^{3}-486y^{12}z^{4}-596y^{11}z^{5}+4020y^{10}z^{6}-104072y^{9}z^{7}-52824y^{8}z^{8}+59472y^{7}z^{9}-34288y^{6}z^{10}-117536y^{5}z^{11}-120096y^{4}z^{12}-55488y^{3}z^{13}-16448y^{2}z^{14}-1920yz^{15}-128z^{16})}$ |
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.1.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-48.ki.4.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kj.4.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kk.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kl.3.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.km.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kn.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ko.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kp.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-48.cg.1.39 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cn.2.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.df.2.30 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.dg.1.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ec.4.19 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ef.3.22 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.en.2.27 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.eo.1.23 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.fn.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fo.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fp.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fq.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fr.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fs.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ft.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fu.3.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gd.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ge.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gf.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gg.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gh.4.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gi.4.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gj.2.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gk.2.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.hf.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hg.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hh.3.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hi.3.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.id.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ie.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.if.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ig.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bbh.3.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbi.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbj.2.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbk.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfx.3.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfy.3.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfz.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bga.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhx.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhy.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhz.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bia.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bib.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bic.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bid.2.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bie.2.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-48.bw.2.1 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-240.cme.3.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmf.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmg.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmh.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmi.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmj.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmk.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cml.3.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.sj.3.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.sk.2.38 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.st.3.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.su.1.38 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tn.4.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.to.3.38 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tx.3.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ty.1.42 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wd.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.we.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wf.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wg.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wh.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wi.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wj.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wk.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xj.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xk.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xl.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xm.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xn.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xo.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xp.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xq.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zj.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zk.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zl.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zm.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbf.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbg.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbh.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbi.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fuj.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuk.3.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ful.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fum.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwf.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwg.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwh.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwi.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ger.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ges.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.get.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.geu.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gev.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gew.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gex.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gey.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |