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: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.5641 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&40\\12&11\end{bmatrix}$, $\begin{bmatrix}7&28\\12&41\end{bmatrix}$, $\begin{bmatrix}19&43\\12&19\end{bmatrix}$, $\begin{bmatrix}25&26\\0&5\end{bmatrix}$, $\begin{bmatrix}25&33\\36&29\end{bmatrix}$, $\begin{bmatrix}31&18\\12&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.qe.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: | $2^{13}\cdot3^{3}$ |
| 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 |
|---|
| $(0:0:1)$, $(0:2:1)$, $(0:1:0)$, $(0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{126x^{2}y^{22}-1260x^{2}y^{21}z+4788x^{2}y^{20}z^{2}-7488x^{2}y^{19}z^{3}-504x^{2}y^{18}z^{4}+17136x^{2}y^{17}z^{5}-474192x^{2}y^{16}z^{6}+1822464x^{2}y^{15}z^{7}+456768x^{2}y^{14}z^{8}-40321152x^{2}y^{13}z^{9}+34867584x^{2}y^{12}z^{10}+237726720x^{2}y^{11}z^{11}-502808832x^{2}y^{10}z^{12}-1391334912x^{2}y^{9}z^{13}+1546689024x^{2}y^{8}z^{14}+4810346496x^{2}y^{7}z^{15}-1218968064x^{2}y^{6}z^{16}-9933364224x^{2}y^{5}z^{17}-5638413312x^{2}y^{4}z^{18}+6442057728x^{2}y^{3}z^{19}+10200508416x^{2}y^{2}z^{20}+5368688640x^{2}yz^{21}+1073737728x^{2}z^{22}+y^{24}-138y^{23}z+1446y^{22}z^{2}-5212y^{21}z^{3}+3804y^{20}z^{4}+20184y^{19}z^{5}+141688y^{18}z^{6}-662832y^{17}z^{7}-528480y^{16}z^{8}+20346176y^{15}z^{9}-17186496y^{14}z^{10}-124073088y^{13}z^{11}+327322240y^{12}z^{12}+772195584y^{11}z^{13}-1266614016y^{10}z^{14}-2749184512y^{9}z^{15}+2912590080y^{8}z^{16}+7930031616y^{7}z^{17}-937820672y^{6}z^{18}-12615011328y^{5}z^{19}-7784223744y^{4}z^{20}+5905627136y^{3}z^{21}+10200569856y^{2}z^{22}+5368713216yz^{23}+1073741824z^{24}}{z^{3}y^{3}(y-2z)(y+z)(2x^{2}y^{14}-12x^{2}y^{13}z+20x^{2}y^{12}z^{2}+96x^{2}y^{11}z^{3}-312x^{2}y^{10}z^{4}-144x^{2}y^{9}z^{5}+1104x^{2}y^{8}z^{6}-1824x^{2}y^{6}z^{8}-320x^{2}y^{5}z^{9}+1728x^{2}y^{4}z^{10}+1024x^{2}y^{3}z^{11}-640x^{2}y^{2}z^{12}-768x^{2}yz^{13}-256x^{2}z^{14}-y^{16}+6y^{15}z-10y^{14}z^{2}+12y^{13}z^{3}-148y^{12}z^{4}+456y^{11}z^{5}+8y^{10}z^{6}-1360y^{9}z^{7}+336y^{8}z^{8}+2208y^{7}z^{9}+32y^{6}z^{10}-2240y^{5}z^{11}-960y^{4}z^{12}+896y^{3}z^{13}+896y^{2}z^{14}+256yz^{15})}$ |
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.1.30 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.1.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kj.1.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kk.3.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kl.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.km.3.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kn.3.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ko.3.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kp.3.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-48.cg.4.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cn.3.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.df.1.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.dg.2.19 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ec.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ef.1.19 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.en.1.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.eo.1.21 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.fn.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fo.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fp.3.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fq.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fr.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fs.1.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ft.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fu.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gd.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ge.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gf.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gg.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gh.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gi.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gj.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gk.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.hf.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hg.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hh.2.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hi.2.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.id.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ie.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.if.2.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ig.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bbh.2.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbi.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbj.3.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbk.3.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfx.2.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfy.2.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfz.3.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bga.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhx.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhy.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhz.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bia.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bib.3.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bic.3.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bid.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bie.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-48.bw.1.1 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-240.cme.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmf.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmg.4.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmh.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmi.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmj.4.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmk.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cml.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.sj.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.sk.3.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.st.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.su.4.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tn.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.to.1.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tx.2.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ty.2.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wd.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.we.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wf.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wg.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wh.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wi.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wj.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wk.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xj.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xk.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xl.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xm.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xn.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xo.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xp.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xq.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zj.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zk.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zl.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zm.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbf.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbg.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbh.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbi.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fuj.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuk.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ful.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fum.3.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwf.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwg.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwh.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwi.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ger.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ges.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.get.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.geu.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gev.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gew.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gex.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gey.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |