Properties

Label 48.192.3-24.gg.1.7
Level $48$
Index $192$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $288$
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 $1^{2}\cdot2\cdot3^{2}\cdot4\cdot6\cdot8^{2}\cdot12\cdot24^{2}$ 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: 24Y3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.3.6456

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}7&47\\12&7\end{bmatrix}$, $\begin{bmatrix}11&39\\12&19\end{bmatrix}$, $\begin{bmatrix}25&38\\24&17\end{bmatrix}$, $\begin{bmatrix}35&1\\0&25\end{bmatrix}$, $\begin{bmatrix}35&13\\24&5\end{bmatrix}$, $\begin{bmatrix}35&36\\12&5\end{bmatrix}$
Contains $-I$: no $\quad$ (see 24.96.3.gg.1 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: $2^{13}\cdot3^{5}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 24.2.a.a, 288.2.d.b

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ 6 x^{4} - 8 x^{3} y - 4 x^{3} z - 3 x^{2} y^{2} - 6 x^{2} y z + 6 x^{2} z^{2} + 2 x y^{3} + \cdots + 2 y z^{3} $
Copy content Toggle raw display

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
$(0:0:1)$, $(0:1:1)$, $(0:2:1)$, $(0:1:0)$

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{1}{3^3}\cdot\frac{2427936093245308513903120x^{3}y^{21}+70190890517443622587191528x^{3}y^{20}z+887940679609409357717786784x^{3}y^{19}z^{2}+6426594017559458420849884928x^{3}y^{18}z^{3}+29100456427834167163004916864x^{3}y^{17}z^{4}+83951367488097716898011689056x^{3}y^{16}z^{5}+146337637447652327785111056384x^{3}y^{15}z^{6}+116886709433252204869358154240x^{3}y^{14}z^{7}-61152439968755621554084674048x^{3}y^{13}z^{8}-214468869335906668957880839424x^{3}y^{12}z^{9}-119775493045818426127763039232x^{3}y^{11}z^{10}+87511953528106379981706491904x^{3}y^{10}z^{11}+109256366957575297467993436160x^{3}y^{9}z^{12}-3514155043540366499151522816x^{3}y^{8}z^{13}-37437439913258618966030254080x^{3}y^{7}z^{14}-4738943777915595803140300800x^{3}y^{6}z^{15}+6236293490543483748333932544x^{3}y^{5}z^{16}+900924389909341433762531328x^{3}y^{4}z^{17}-480032257835918272737566720x^{3}y^{3}z^{18}-43556228911816637930766336x^{3}y^{2}z^{19}+11564059442982295330652160x^{3}yz^{20}+243474531221087389933568x^{3}z^{21}+448497278848370624077434x^{2}y^{22}+15396284907704214305799996x^{2}y^{21}z+229170492951794300489865756x^{2}y^{20}z^{2}+1938666367520240948299373664x^{2}y^{19}z^{3}+10211307177851113012234414680x^{2}y^{18}z^{4}+34146118563204414147605679888x^{2}y^{17}z^{5}+68776280922711714727931446608x^{2}y^{16}z^{6}+62971068852639053059479148800x^{2}y^{15}z^{7}-40387467195717787501272599616x^{2}y^{14}z^{8}-159066660633168900231942069120x^{2}y^{13}z^{9}-102403030906075381592816360832x^{2}y^{12}z^{10}+88120316292310957629469507584x^{2}y^{11}z^{11}+127998952388592234142859546880x^{2}y^{10}z^{12}-4902861120254102688504340992x^{2}y^{9}z^{13}-61156445463744882760229773824x^{2}y^{8}z^{14}-9282785500713832908660707328x^{2}y^{7}z^{15}+14896121524455184248609686016x^{2}y^{6}z^{16}+2693037694496817203013897216x^{2}y^{5}z^{17}-1855789185764650643720696832x^{2}y^{4}z^{18}-234360636056463269201141760x^{2}y^{3}z^{19}+95007772734985983748356096x^{2}y^{2}z^{20}+4298440643413022829563904x^{2}yz^{21}-840041628431973300989952x^{2}z^{22}-533123608896787445113444xy^{23}-13402998884752016909939812xy^{22}z-139108272315466025620181632xy^{21}z^{2}-736343842237823633964941384xy^{20}z^{3}-1776207815541807275339532464xy^{19}z^{4}+945442201270979227047863344xy^{18}z^{5}+17852023914363050784046647360xy^{17}z^{6}+44442766259866724323736777568xy^{16}z^{7}+30703917282778167576740221056xy^{15}z^{8}-57181105220703350627725244800xy^{14}z^{9}-114855714412231464230657655808xy^{13}z^{10}-21183927110797941409025733376xy^{12}z^{11}+98729528252395969919287031296xy^{11}z^{12}+60539249758148598282080510464xy^{10}z^{13}-36064458127130866029602772992xy^{9}z^{14}-34806902133901065658823998464xy^{8}z^{15}+5931151285386519089471032320xy^{7}z^{16}+9052088445116695742463962112xy^{6}z^{17}-393149704226083361630371840xy^{5}z^{18}-1108836677424257766110537728xy^{4}z^{19}+12006873826766740546555904xy^{3}z^{20}+52519893175399100435820544xy^{2}z^{21}-581956501856527614328832xyz^{22}-422770823238933289115648xz^{23}+14348907y^{24}-266561804448394066930490y^{23}z-6613702408751794239503774y^{22}z^{2}-68005364988321613238028996y^{21}z^{3}-362232002500644618400592716y^{20}z^{4}-952984232414345702907345736y^{19}z^{5}-353726100359807269059085464y^{18}z^{6}+4875482621361302066698687920y^{17}z^{7}+12075517430318079890445191520y^{16}z^{8}+4962642604322095036643932224y^{15}z^{9}-19926707917532287077697299776y^{14}z^{10}-24912796506596669439564311936y^{13}z^{11}+8232502885987509485535302016y^{12}z^{12}+25895861357092829115888372992y^{11}z^{13}+2183059416131842877769608960y^{10}z^{14}-12754401976070204460971908608y^{9}z^{15}-2336352275610898220241622272y^{8}z^{16}+3394671702635036962056261120y^{7}z^{17}+528631421489720513266383360y^{6}z^{18}-464851189032385347738776576y^{5}z^{19}-34337509980420721343396864y^{4}z^{20}+25039713451398260016961536y^{3}z^{21}+77571984058661329750016y^{2}z^{22}-211385477430318565691392yz^{23}+2742118830047232z^{24}}{(y-z)^{3}(716694953818103740x^{3}y^{18}+11770634434921478136x^{3}y^{17}z+68441208897409725816x^{3}y^{16}z^{2}+130887328833736008384x^{3}y^{15}z^{3}-211096983196850283072x^{3}y^{14}z^{4}-1011984900721696271520x^{3}y^{13}z^{5}-111892222345937708640x^{3}y^{12}z^{6}+2863688368588271093376x^{3}y^{11}z^{7}+1138366864478172622464x^{3}y^{10}z^{8}-4457611347416098933376x^{3}y^{9}z^{9}-1507785289199845225344x^{3}y^{8}z^{10}+3998661070295900620800x^{3}y^{7}z^{11}+620215673470360894464x^{3}y^{6}z^{12}-1863412237884096400896x^{3}y^{5}z^{13}+23025127166028928512x^{3}y^{4}z^{14}+360901356640044312576x^{3}y^{3}z^{15}-36263349884534510592x^{3}y^{2}z^{16}-18096910634730147840x^{3}yz^{17}+1468913554569650176x^{3}z^{18}+132390522868884888x^{2}y^{19}+2891727087709615518x^{2}y^{18}z+22915448100261990744x^{2}y^{17}z^{2}+71006600909100093612x^{2}y^{16}z^{3}-10388488027052606808x^{2}y^{15}z^{4}-499016730517429685592x^{2}y^{14}z^{5}-511542947283052700352x^{2}y^{13}z^{6}+1489491147705010333872x^{2}y^{12}z^{7}+1947169461088176764064x^{2}y^{11}z^{8}-2814350077815600908448x^{2}y^{10}z^{9}-3078553258843843860864x^{2}y^{9}z^{10}+3577803150947499559488x^{2}y^{8}z^{11}+2178402053809249941888x^{2}y^{7}z^{12}-2646507310876458826368x^{2}y^{6}z^{13}-506934966283608615936x^{2}y^{5}z^{14}+905799414398093757696x^{2}y^{4}z^{15}-39281459317776462336x^{2}y^{3}z^{16}-102691994589604469760x^{2}y^{2}z^{17}+12117212224355328000x^{2}yz^{18}+1538626703988793344x^{2}z^{19}-157371109851158350xy^{20}-1991422459409157832xy^{19}z-5944017821611865188xy^{18}z^{2}+18376943445683314248xy^{17}z^{3}+108306954132994384680xy^{16}z^{4}-8901399223982959872xy^{15}z^{5}-615040251414201351792xy^{14}z^{6}-218688359500708836192xy^{13}z^{7}+1848004461250957667232xy^{12}z^{8}+582423937163235286400xy^{11}z^{9}-3276102296255005615936xy^{10}z^{10}-387430188303797433472xy^{9}z^{11}+3335109271628278022016xy^{8}z^{12}-263388578892440798208xy^{7}z^{13}-1784137075053742002432xy^{6}z^{14}+381068681500944373248xy^{5}z^{15}+435245781507498003456xy^{4}z^{16}-120113526503142383616xy^{3}z^{17}-36413650631836456960xy^{2}z^{18}+9378138406013444096xyz^{19}+393968117853863936xz^{20}-78685554925579175y^{20}z-969794694227217176y^{19}z^{2}-2838431028257070582y^{18}z^{3}+7585368296730764400y^{17}z^{4}+41083721822476655316y^{16}z^{5}-19719800676508800288y^{15}z^{6}-222308536483408331112y^{14}z^{7}+47379816260192174880y^{13}z^{8}+638122253435035986480y^{12}z^{9}-218356511849077153856y^{11}z^{10}-983584869137467258784y^{10}z^{11}+557609921751862677504y^{9}z^{12}+710559565738914329280y^{8}z^{13}-582768054743441587200y^{7}z^{14}-159865868075938306944y^{6}z^{15}+225274938036727478784y^{5}z^{16}-17560229596969939968y^{4}z^{17}-23918462043847498752y^{3}z^{18}+4156674482814324736y^{2}z^{19}+196984058926931968yz^{20})}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
48.48.0-24.by.1.5 $48$ $4$ $4$ $0$ $0$ full Jacobian
48.96.1-24.ir.1.8 $48$ $2$ $2$ $1$ $0$ $2$
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
48.384.5-24.ga.3.15 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.ga.4.15 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.ge.1.16 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.ge.2.16 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.gi.3.15 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.gi.4.15 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.gm.3.16 $48$ $2$ $2$ $5$ $0$ $2$
48.384.5-24.gm.4.16 $48$ $2$ $2$ $5$ $0$ $2$
48.384.7-24.dk.1.8 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-24.dt.2.4 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-24.dz.2.9 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-24.ec.1.6 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-24.eh.1.2 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-24.em.2.3 $48$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
48.384.7-24.eq.2.5 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.eq.3.17 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eq.4.17 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.er.2.5 $48$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
48.384.7-48.eu.1.17 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eu.2.17 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.ev.1.3 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.ev.2.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.ez.3.7 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.ez.4.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.fd.1.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.fd.2.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.fh.1.4 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-24.fh.2.4 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fo.3.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fo.4.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.3.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.4.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.gp.2.7 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.gv.2.19 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.gx.1.21 $48$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
48.384.7-48.hd.1.13 $48$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
48.384.7-48.hp.2.4 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.hr.2.18 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.hx.1.19 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.7-48.hz.1.7 $48$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
48.384.9-48.bat.1.7 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bav.1.19 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bbb.2.18 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.bbd.2.4 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.bfh.1.13 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bfn.1.21 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bfp.2.19 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bfv.2.7 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bha.3.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bha.4.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bhe.3.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bhe.4.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bhy.1.17 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bhy.2.17 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bic.3.17 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.384.9-48.bic.4.17 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.576.13-24.ln.1.4 $48$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
240.384.5-120.beu.3.10 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.beu.4.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bey.3.9 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bey.4.2 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bfk.1.6 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bfk.2.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bfo.1.5 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.bfo.2.2 $240$ $2$ $2$ $5$ $?$ not computed
240.384.7-120.lg.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.lm.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ls.2.21 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ly.1.21 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.mk.1.31 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.mq.2.10 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.mw.2.22 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.nc.2.22 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.nl.1.19 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.nl.2.18 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.np.1.11 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.np.2.10 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ob.1.21 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ob.2.18 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.of.3.13 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.of.4.10 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ui.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ui.2.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.um.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.um.2.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.vw.3.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.vw.4.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.wa.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.wa.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.yd.2.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.yj.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.yt.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.yz.2.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bab.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bad.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bar.1.33 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bat.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.9-240.ftf.2.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fth.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ftv.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ftx.2.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fuz.2.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fvf.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fvp.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fvv.2.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gcw.1.2 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gcw.2.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gda.3.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gda.4.5 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gek.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gek.2.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.geo.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.geo.2.33 $240$ $2$ $2$ $9$ $?$ not computed