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: | 48.192.3.5593 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&23\\24&35\end{bmatrix}$, $\begin{bmatrix}19&9\\36&19\end{bmatrix}$, $\begin{bmatrix}35&5\\12&31\end{bmatrix}$, $\begin{bmatrix}35&17\\24&37\end{bmatrix}$, $\begin{bmatrix}43&4\\24&11\end{bmatrix}$, $\begin{bmatrix}43&11\\12&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.pz.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: | $2^{9}\cdot3^{3}$ |
| 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 + x^{3} z + 2 x^{2} y z - 2 x y^{2} z - 2 x y z^{2} + y^{3} z - 2 y^{2} z^{2} + 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 |
|---|
| $(1:1:1)$, $(0:0:1)$, $(0:1:0)$, $(1:0:0)$, $(-2:1:1)$, $(0:1:1)$ |
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{16777216x^{24}+201326592x^{22}z^{2}-402653184x^{21}z^{3}+1811939328x^{20}z^{4}-5637144576x^{19}z^{5}+17985175552x^{18}z^{6}-55566139392x^{17}z^{7}+167906377728x^{16}z^{8}-501437431808x^{15}z^{9}+1484984942592x^{14}z^{10}-4377645416448x^{13}z^{11}+12881412227072x^{12}z^{12}-37910602579968x^{11}z^{13}+111755585912832x^{10}z^{14}-330310902349824x^{9}z^{15}+979475613351936x^{8}z^{16}-2914970681475072x^{7}z^{17}+8708004316708864x^{6}z^{18}-26113946620526592x^{5}z^{19}+78612184430542848x^{4}z^{20}-237542001053007872x^{3}z^{21}-660x^{2}y^{22}+1670072x^{2}y^{21}z-238544796x^{2}y^{20}z^{2}-6787445456x^{2}y^{19}z^{3}+170983821684x^{2}y^{18}z^{4}+457269999192x^{2}y^{17}z^{5}-5272836441828x^{2}y^{16}z^{6}-22539176734656x^{2}y^{15}z^{7}+4505622388408x^{2}y^{14}z^{8}+201775872617968x^{2}y^{13}z^{9}+542413285245160x^{2}y^{12}z^{10}+697919714690336x^{2}y^{11}z^{11}+380390073188584x^{2}y^{10}z^{12}-365255910347280x^{2}y^{9}z^{13}-1473849836855624x^{2}y^{8}z^{14}-3665924500451264x^{2}y^{7}z^{15}-7060674602272484x^{2}y^{6}z^{16}-20006339524004264x^{2}y^{5}z^{17}-23718946701176460x^{2}y^{4}z^{18}-122386057127801552x^{2}y^{3}z^{19}-52237861830256540x^{2}y^{2}z^{20}-658422317585957960x^{2}yz^{21}-660x^{2}z^{22}-12xy^{23}-137372xy^{22}z+141326780xy^{21}z^{2}-6541359380xy^{20}z^{3}-1900954660xy^{19}z^{4}+1002024713964xy^{18}z^{5}+584845708980xy^{17}z^{6}-26364156911484xy^{16}z^{7}-70368068590904xy^{15}z^{8}+94204139067944xy^{14}z^{9}+724746438833944xy^{13}z^{10}+1454409000401400xy^{12}z^{11}+1535668842117624xy^{11}z^{12}+1086424695306008xy^{10}z^{13}+1127152229472808xy^{9}z^{14}+2397527991427784xy^{8}z^{15}+5726489723723908xy^{7}z^{16}+11679263556456116xy^{6}z^{17}+28462693599133420xy^{5}z^{18}+51724160932625372xy^{4}z^{19}+139898731072505580xy^{3}z^{20}+273042852243339708xy^{2}z^{21}+553696186536421220xyz^{22}-12xz^{23}+y^{24}-7040y^{23}z-9981884y^{22}z^{2}+1744867584y^{21}z^{3}-37032341278y^{20}z^{4}-135283108096y^{19}z^{5}+2085994948916y^{18}z^{6}+6300626956416y^{17}z^{7}-17213808760465y^{16}z^{8}-85587347646720y^{15}z^{9}-84627384900984y^{14}z^{10}+74586139028480y^{13}z^{11}+176091011224412y^{12}z^{12}+14679666748416y^{11}z^{13}-203218948136312y^{10}z^{14}-395358508427520y^{9}z^{15}-179102668871313y^{8}z^{16}-2949170119724928y^{7}z^{17}+5731437613214004y^{6}z^{18}-36720979308734720y^{5}z^{19}+91733133169479906y^{4}z^{20}-314026621222743808y^{3}z^{21}+494465736730193988y^{2}z^{22}-237542001053014912yz^{23}+z^{24}}{16777216x^{8}z^{16}-268435456x^{7}z^{17}+2483027968x^{6}z^{18}-17314086912x^{5}z^{19}+100931731456x^{4}z^{20}-519691042816x^{3}z^{21}-x^{2}y^{22}-412x^{2}y^{21}z-21556x^{2}y^{20}z^{2}-370338x^{2}y^{19}z^{3}-2637432x^{2}y^{18}z^{4}-7111932x^{2}y^{17}z^{5}+4380359x^{2}y^{16}z^{6}+54663560x^{2}y^{15}z^{7}+42483858x^{2}y^{14}z^{8}-188038888x^{2}y^{13}z^{9}-123897004x^{2}y^{12}z^{10}+122332468x^{2}y^{11}z^{11}+1218280276x^{2}y^{10}z^{12}-3409264360x^{2}y^{9}z^{13}+7290241170x^{2}y^{8}z^{14}-17125205624x^{2}y^{7}z^{15}+40001263303x^{2}y^{6}z^{16}-86980199676x^{2}y^{5}z^{17}+174732067208x^{2}y^{4}z^{18}-327927834274x^{2}y^{3}z^{19}+528129960908x^{2}y^{2}z^{20}-1258559635868x^{2}yz^{21}-x^{2}z^{22}+83xy^{22}z+7285xy^{21}z^{2}+169308xy^{20}z^{3}+1514668xy^{19}z^{4}+5309593xy^{18}z^{5}+1330607xy^{17}z^{6}-32734624xy^{16}z^{7}-46123456xy^{15}z^{8}+77701870xy^{14}z^{9}+182285826xy^{13}z^{10}-172683944xy^{12}z^{11}-172683944xy^{11}z^{12}-86149630xy^{10}z^{13}+883008238xy^{9}z^{14}-1656736192xy^{8}z^{15}+2651619936xy^{7}z^{16}-4025201233xy^{6}z^{17}+5642454169xy^{5}z^{18}-6977807188xy^{4}z^{19}+24964666716xy^{3}z^{20}+215016807541xy^{2}z^{21}+1140313817171xyz^{22}-10y^{23}z-1489y^{22}z^{2}-43038y^{21}z^{3}-408554y^{20}z^{4}-1103318y^{19}z^{5}+2748163y^{18}z^{6}+14153294y^{17}z^{7}-8885752y^{16}z^{8}-78954020y^{15}z^{9}+68829390y^{14}z^{10}+65947092y^{13}z^{11}+479416260y^{12}z^{12}-2618407468y^{11}z^{13}+7048151246y^{10}z^{14}-16721952292y^{9}z^{15}+39719561736y^{8}z^{16}-90717030834y^{7}z^{17}+192739405571y^{6}z^{18}-379635946966y^{5}z^{19}+693099938838y^{4}z^{20}-1243862837278y^{3}z^{21}+1320098462255y^{2}z^{22}-519691042826yz^{23}}$ |
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.1.17 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kb.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kc.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kd.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ke.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kf.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kg.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kh.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-48.cg.1.38 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cn.1.20 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cw.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cx.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ec.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ef.1.23 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.eh.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ei.1.11 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.ep.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eq.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.er.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.es.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.et.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eu.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ev.4.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ew.4.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fv.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fw.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fx.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fy.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fz.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ga.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gb.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gc.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gl.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gm.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gn.3.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.go.3.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hj.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hk.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hl.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hm.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.ban.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bao.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bap.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.baq.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfd.3.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfe.3.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bff.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfg.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bgz.4.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bha.4.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhb.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhc.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhd.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhe.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhf.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhg.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-48.cf.2.17 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-240.clg.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clh.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cli.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clj.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clk.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cll.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clm.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cln.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.rx.2.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ry.1.38 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rz.3.18 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.sa.1.18 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tb.3.34 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tc.1.42 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.td.3.18 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.te.1.18 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tz.4.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ua.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ub.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uc.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ud.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ue.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uf.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ug.4.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wl.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wm.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wn.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wo.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wp.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wq.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.wr.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ws.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xr.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xs.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xt.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xu.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zn.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zo.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zp.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zq.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fsr.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fss.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fst.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsu.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fun.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuo.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fup.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuq.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcn.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gco.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcp.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcq.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcr.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcs.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gct.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcu.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |