Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $24$ | ||
| 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 $6$ are rational) | Cusp widths | $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}\cdot16\cdot48$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| 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: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.5882 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&43\\36&17\end{bmatrix}$, $\begin{bmatrix}11&39\\24&41\end{bmatrix}$, $\begin{bmatrix}13&38\\12&47\end{bmatrix}$, $\begin{bmatrix}19&19\\12&35\end{bmatrix}$, $\begin{bmatrix}25&3\\36&1\end{bmatrix}$, $\begin{bmatrix}35&40\\36&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.pz.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: | $2^{9}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 24.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y + 2 x^{2} y^{2} + 2 x^{2} y z + x y^{3} - 2 x y^{2} z + 2 x y z^{2} + x z^{3} - y z^{3} $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:1:0)$, $(0:1:0)$, $(1:0:0)$, $(1/2:-1/2:1)$, $(0:0:1)$, $(-1:1:1)$ |
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{x^{24}+12x^{23}z+60x^{22}z^{2}+880x^{21}z^{3}+6732x^{20}z^{4}+21888x^{19}z^{5}+211996x^{18}z^{6}+1106136x^{17}z^{7}+1656804x^{16}z^{8}+17331240x^{15}z^{9}+54081780x^{14}z^{10}-50785248x^{13}z^{11}+457941814x^{12}z^{12}+334614000x^{11}z^{13}-3826338672x^{10}z^{14}+17364153408x^{9}z^{15}-31504106808x^{8}z^{16}-34666199376x^{7}z^{17}+485901365136x^{6}z^{18}-1831572209616x^{5}z^{19}+3392705429580x^{4}z^{20}+2884259619688x^{3}z^{21}+48378511622122x^{2}y^{22}+3663572743742296x^{2}y^{21}z+75259371897406284x^{2}y^{20}z^{2}+646484249809681386x^{2}y^{19}z^{3}+2697975034839378308x^{2}y^{18}z^{4}+5425321321809271756x^{2}y^{17}z^{5}+3202894955943345944x^{2}y^{16}z^{6}-6263474610847902744x^{2}y^{15}z^{7}-10933461133991848196x^{2}y^{14}z^{8}-1623191080848732196x^{2}y^{13}z^{9}+8076995588471315164x^{2}y^{12}z^{10}+5029192724412038412x^{2}y^{11}z^{11}-2016878574791178526x^{2}y^{10}z^{12}-2734974707002388892x^{2}y^{9}z^{13}-200466566455304092x^{2}y^{8}z^{14}+647512220032108074x^{2}y^{7}z^{15}+202450886592456444x^{2}y^{6}z^{16}-61222219917077792x^{2}y^{5}z^{17}-39256225235508260x^{2}y^{4}z^{18}-927414561054658x^{2}y^{3}z^{19}+2872973805972670x^{2}y^{2}z^{20}+371216896303639x^{2}yz^{21}-46495416422532x^{2}z^{22}+43980465111020xy^{23}+2669614232233784xy^{22}z+41398811808560920xy^{21}z^{2}+241182273586000874xy^{20}z^{3}+532695791300179340xy^{19}z^{4}+127291559707380068xy^{18}z^{5}-466457912429786912xy^{17}z^{6}+1987953332024199768xy^{16}z^{7}+4965000423420826872xy^{15}z^{8}-77019545814457620xy^{14}z^{9}-7737794517357484568xy^{13}z^{10}-4442318964652377716xy^{12}z^{11}+4006611551496673740xy^{11}z^{12}+4322497298622858740xy^{10}z^{13}-329858816339494416xy^{9}z^{14}-1695694833838987646xy^{8}z^{15}-392199472008809428xy^{7}z^{16}+291884224204142736xy^{6}z^{17}+144144226130755876xy^{5}z^{18}-11129741227814654xy^{4}z^{19}-19307332938167212xy^{3}z^{20}-2530368540955519xy^{2}z^{21}+826897479450218xyz^{22}+182086747708992xz^{23}+y^{24}-12y^{23}z+60y^{22}z^{2}-43980465111900y^{21}z^{3}-2757575162449092y^{20}z^{4}-46826001203272416y^{19}z^{5}-329235562783667348y^{18}z^{6}-1092738635951496056y^{17}z^{7}-1579605682141504848y^{16}z^{8}-20781867124523200y^{15}z^{9}+2623743682958068624y^{14}z^{10}+2267529835791723528y^{13}z^{11}-918432338816528188y^{12}z^{12}-2066633572238725080y^{11}z^{13}-391349599286699256y^{10}z^{14}+732970037958985884y^{9}z^{15}+362310810931646492y^{8}z^{16}-92530201511954160y^{7}z^{17}-99346875669531332y^{6}z^{18}-7225053925261688y^{5}z^{19}+11579451549170424y^{4}z^{20}+2895733687308228y^{3}z^{21}-416228567560550y^{2}z^{22}-182086747708992yz^{23}+4096z^{24}}{z^{3}(x^{21}+9x^{20}z+30x^{19}z^{2}+21x^{18}z^{3}-102x^{17}z^{4}-156x^{16}z^{5}+229x^{15}z^{6}+312x^{14}z^{7}-648x^{13}z^{8}+124x^{12}z^{9}+1188x^{11}z^{10}-2340x^{10}z^{11}+1396x^{9}z^{12}+2493x^{8}z^{13}-7602x^{7}z^{14}+10893x^{6}z^{15}-9918x^{5}z^{16}+4068x^{4}z^{17}+4471x^{3}z^{18}+19x^{2}y^{19}+4776x^{2}y^{18}z+290428x^{2}y^{17}z^{2}+7032968x^{2}y^{16}z^{3}+80514200x^{2}y^{15}z^{4}+464546000x^{2}y^{14}z^{5}+1304034122x^{2}y^{13}z^{6}+1254329864x^{2}y^{12}z^{7}-1433838866x^{2}y^{11}z^{8}-3260434731x^{2}y^{10}z^{9}-51350221x^{2}y^{9}z^{10}+2503842600x^{2}y^{8}z^{11}+465476924x^{2}y^{7}z^{12}-853015343x^{2}y^{6}z^{13}-144755642x^{2}y^{5}z^{14}+127378359x^{2}y^{4}z^{15}+10284835x^{2}y^{3}z^{16}-6407864x^{2}y^{2}z^{17}+245260x^{2}yz^{18}-9492x^{2}z^{19}+17xy^{20}+4030xy^{19}z+210294xy^{18}z^{2}+4171443xy^{17}z^{3}+36010808xy^{16}z^{4}+131147936xy^{15}z^{5}+114939981xy^{14}z^{6}-248003322xy^{13}z^{7}+155825156xy^{12}z^{8}+1683168981xy^{11}z^{9}+334540251xy^{10}z^{10}-2695411432xy^{9}z^{11}-872871382xy^{8}z^{12}+1688784570xy^{7}z^{13}+498604036xy^{6}z^{14}-455997212xy^{5}z^{15}-101757835xy^{4}z^{16}+48200284xy^{3}z^{17}+6308115xy^{2}z^{18}-1457704xyz^{19}+19992xz^{20}-y^{21}+9y^{20}z-30y^{19}z^{2}+4y^{18}z^{3}-3962y^{17}z^{4}-218544y^{16}z^{5}-4600288y^{15}z^{6}-44766358y^{14}z^{7}-211100844y^{13}z^{8}-440089820y^{12}z^{9}-141478394y^{11}z^{10}+741032064y^{10}z^{11}+645188374y^{9}z^{12}-442121364y^{8}z^{13}-468675240y^{7}z^{14}+133532122y^{6}z^{15}+121716442y^{5}z^{16}-22067736y^{4}z^{17}-9591190y^{3}z^{18}+1507180y^{2}z^{19}-19992yz^{20})}$ |
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.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.ka.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kb.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kc.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kd.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ke.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kf.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kg.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kh.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-48.cg.4.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cn.4.19 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cw.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cx.2.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ec.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ef.3.21 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.eh.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ei.1.27 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.ep.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eq.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.er.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.es.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.et.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eu.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ev.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ew.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fv.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fw.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fx.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fy.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fz.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ga.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gb.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gc.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gl.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gm.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gn.2.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.go.2.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hj.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hk.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hl.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hm.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.ban.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bao.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bap.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.baq.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfd.2.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfe.2.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bff.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfg.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bgz.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bha.1.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhb.3.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhc.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhd.3.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhe.3.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhf.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhg.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-48.cf.1.17 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-240.clg.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clh.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cli.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clj.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clk.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cll.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clm.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cln.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.rx.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ry.4.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rz.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.sa.4.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tb.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tc.3.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.td.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.te.2.50 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tz.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ua.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ub.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uc.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ud.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ue.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uf.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ug.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wl.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wm.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wn.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wo.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wp.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wq.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wr.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ws.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xr.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xs.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xt.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xu.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zn.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zo.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zp.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zq.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fsr.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fss.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fst.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsu.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fun.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuo.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fup.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuq.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcn.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gco.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcp.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcq.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcr.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcs.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gct.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcu.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |