↑

Properties

Label 48.48.1.bs.2
Level $48$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.64

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&39\\44&37\end{bmatrix}$, $\begin{bmatrix}9&28\\40&5\end{bmatrix}$, $\begin{bmatrix}19&46\\40&23\end{bmatrix}$, $\begin{bmatrix}27&20\\44&17\end{bmatrix}$, $\begin{bmatrix}47&8\\32&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bs.2.1, 48.96.1-48.bs.2.2, 48.96.1-48.bs.2.3, 48.96.1-48.bs.2.4, 48.96.1-48.bs.2.5, 48.96.1-48.bs.2.6, 48.96.1-48.bs.2.7, 48.96.1-48.bs.2.8, 48.96.1-48.bs.2.9, 48.96.1-48.bs.2.10, 48.96.1-48.bs.2.11, 48.96.1-48.bs.2.12, 48.96.1-48.bs.2.13, 48.96.1-48.bs.2.14, 48.96.1-48.bs.2.15, 48.96.1-48.bs.2.16, 240.96.1-48.bs.2.1, 240.96.1-48.bs.2.2, 240.96.1-48.bs.2.3, 240.96.1-48.bs.2.4, 240.96.1-48.bs.2.5, 240.96.1-48.bs.2.6, 240.96.1-48.bs.2.7, 240.96.1-48.bs.2.8, 240.96.1-48.bs.2.9, 240.96.1-48.bs.2.10, 240.96.1-48.bs.2.11, 240.96.1-48.bs.2.12, 240.96.1-48.bs.2.13, 240.96.1-48.bs.2.14, 240.96.1-48.bs.2.15, 240.96.1-48.bs.2.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 396x - 3024 $
 Copy content Toggle raw display

Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^{11}}{3}\cdot\frac{712x^{2}y^{30}+1480896x^{2}y^{29}z+2343565440x^{2}y^{28}z^{2}+1637232694272x^{2}y^{27}z^{3}+150898643822592x^{2}y^{26}z^{4}-248049189184204800x^{2}y^{25}z^{5}-28732239066209052672x^{2}y^{24}z^{6}+32672587199905190117376x^{2}y^{23}z^{7}+1751049996962233647955968x^{2}y^{22}z^{8}-4009015995325047618081914880x^{2}y^{21}z^{9}-992505369799840748428574392320x^{2}y^{20}z^{10}-286768834624388884080818456100864x^{2}y^{19}z^{11}-129156271314102487354350506052943872x^{2}y^{18}z^{12}-29316071150934884909474122327419518976x^{2}y^{17}z^{13}-5323481788675698098342252587556970430464x^{2}y^{16}z^{14}-1250484670129963636569163776580985259294720x^{2}y^{15}z^{15}-191164368512701121122372569097895994753810432x^{2}y^{14}z^{16}-24366459916969783738925485806915859164042362880x^{2}y^{13}z^{17}-4231241074335050522357735429341053160023260135424x^{2}y^{12}z^{18}-363236412308967324584130798353501579643742914084864x^{2}y^{11}z^{19}-39831138107897990852148442326527826011101150530502656x^{2}y^{10}z^{20}-5337776526380448489261034170529164334073553714690392064x^{2}y^{9}z^{21}-14040879931344616818251848460862002262409985844030996480x^{2}y^{8}z^{22}-54731691611224710450070212052797038294796789543926797172736x^{2}y^{7}z^{23}+2250090928205301273894484832295900417634364814209397194489856x^{2}y^{6}z^{24}-299844998050354273956900241974632894211709037181806512617553920x^{2}y^{5}z^{25}+14094027338145703900276323185307972695124294864096951089441538048x^{2}y^{4}z^{26}-726789689740743900892253774260779883486756975155331645237540945920x^{2}y^{3}z^{27}+27319412795079214088955774535453501278735450777533193665835719196672x^{2}y^{2}z^{28}-460582574635864543691260195471728590591291126726749108872761730662400x^{2}yz^{29}+2722991523843226143030388100599169231331208926602114196206416853729280x^{2}z^{30}-56xy^{31}-236304xy^{30}z-291389184xy^{29}z^{2}-105770499840xy^{28}z^{3}+61077324810240xy^{27}z^{4}+50670061709031936xy^{26}z^{5}+3289067867947253760xy^{25}z^{6}-3423559567621365817344xy^{24}z^{7}+146951417649300101332992xy^{23}z^{8}+86597065188281167648653312xy^{22}z^{9}-179842262292013230363368226816xy^{21}z^{10}-66965863710805335183729269145600xy^{20}z^{11}-14938916388558440473201747490242560xy^{19}z^{12}-5137103308559745413349599046659997696xy^{18}z^{13}-1245087168323570130767310602224439132160xy^{17}z^{14}-204102320896305407530416814560467245596672xy^{16}z^{15}-42222679468676402015712351504648910286094336xy^{15}z^{16}-6733240034168265697984141160503098641731289088xy^{14}z^{17}-756828834428166523237195893033357432356710907904xy^{13}z^{18}-126284031096034488856541335117778303523421440442368xy^{12}z^{19}-11394194996275271970078499322337341329039882966794240xy^{11}z^{20}-979765179197277487603282482331683584411225026914680832xy^{10}z^{21}-158760187148610572551334916162477615912961025703218774016xy^{9}z^{22}+1087728430739725699835915315963443365945729080695480909824xy^{8}z^{23}-1481511428498456690886282528990413144505464838283591176683520xy^{7}z^{24}+62443869874121642238528505212583065239351608357546227318915072xy^{6}z^{25}-7480949620819050944445930511129269340738606701309029408695123968xy^{5}z^{26}+346652194373977479987147918092946381522939625769101692858934493184xy^{4}z^{27}-17082810091138822456549096663625145187150273618435290105729628766208xy^{3}z^{28}+629836412716351456655114312526268652609177695611064901116973996310528xy^{2}z^{29}-10579840931528282391476725537749474297913382278303479793047569373855744xyz^{30}+62548647662012214003559126600593881339365848550327963762524913045340160xz^{31}+y^{32}+8928y^{31}z+4831488y^{30}z^{2}-11042625024y^{29}z^{3}-14714794160640y^{28}z^{4}-4165061072707584y^{27}z^{5}+1696236758991065088y^{26}z^{6}+850215445462718889984y^{25}z^{7}-78025435169012032094208y^{24}z^{8}-64696289821163771388493824y^{23}z^{9}-7157661828912965662518804480y^{22}z^{10}-6251657265715856926856540848128y^{21}z^{11}-3090447727655091563201513409478656y^{20}z^{12}-646846021894120772764202993417453568y^{19}z^{13}-148932359970041554114568077321697230848y^{18}z^{14}-37521972924078141475000097452187493335040y^{17}z^{15}-5641792673572261831342773614661136231170048y^{16}z^{16}-893192038146404138344457905809461058641854464y^{15}z^{17}-152629379041957543325180918052255355584909410304y^{14}z^{18}-13932456283832080606300713278819241629272057577472y^{13}z^{19}-1976607291529767711745774383466514694790877036937216y^{12}z^{20}-211121598750522063915450416532405221780272967363592192y^{11}z^{21}-7396859700187531021175380698704649311459390449067753472y^{10}z^{22}-2529286777667668679820754123109950516208627085636079190016y^{9}z^{23}+71068127225039617821404630551642587620418827509114601472000y^{8}z^{24}-17993453319048090159100021019558658731475287912925289330507776y^{7}z^{25}+809654998884494184653870196742983597436623445620302362259226624y^{6}z^{26}-67059312761138734276612113979264495649034703826760159519760711680y^{5}z^{27}+2909622185564637147753417858728001560237394785402393952889859997696y^{4}z^{28}-113506585208011710447702370735857830004566691434296815427078431178752y^{3}z^{29}+3701906757611066579639800989815093675845099476257045701705566712233984y^{2}z^{30}-60634200430774894406179238305064739604080453650567512369898061815611392yz^{31}+358472992510722003446333632720846229212953804527788360983786053399740416z^{32}}{160x^{2}y^{30}-101290176x^{2}y^{28}z^{2}+16253730625536x^{2}y^{26}z^{4}+1652208227383744512x^{2}y^{24}z^{6}-523104997342961844486144x^{2}y^{22}z^{8}+17565955246077314295095820288x^{2}y^{20}z^{10}+2290341284752517224731100770729984x^{2}y^{18}z^{12}-82478952985171347994957076313793363968x^{2}y^{16}z^{14}-5488464620378883323453941097365823740182528x^{2}y^{14}z^{16}+45284808323035864529415806586542317080894504960x^{2}y^{12}z^{18}+6984850732612550877785758063177655178216773064327168x^{2}y^{10}z^{20}+168795997658122990601229112970873123852206734169891405824x^{2}y^{8}z^{22}+1980331240521081486381532759144883363174036194151821788315648x^{2}y^{6}z^{24}+12678489351103604547207669813101193835519838216789902818451390464x^{2}y^{4}z^{26}+42556796057114007792723686468772305095169205225063741903486572298240x^{2}y^{2}z^{28}+58735606192737821374867204633218371377990488732663836653184114019532800x^{2}z^{30}+7264xy^{30}z-6285233664xy^{28}z^{3}+1625650036830720xy^{26}z^{5}-67490555902214529024xy^{24}z^{7}-18544110648334255303557120xy^{22}z^{9}+1458249997310381007107317039104xy^{20}z^{11}+55074115509107012368753095154335744xy^{18}z^{13}-4620631339328776962087162366115983654912xy^{16}z^{15}-153396226874281005110882688715636499986513920xy^{14}z^{17}+3786002876881339060044718492325744509466129203200xy^{12}z^{19}+252568494575627083028840911031307829909164294502612992xy^{10}z^{21}+5151696974661131613188309310395390583028405384795895365632xy^{8}z^{23}+54624402491088915987491822526705247222808994484557581905821696xy^{6}z^{25}+324665722491056232227494566946045462035814723401259829131418271744xy^{4}z^{27}+1027038442554684815625514888500306331127252669953858608723796558872576xy^{2}z^{29}+1349189927620126649339766923634558875359352593330961029452953334338027520xz^{31}+y^{32}-596736y^{30}z^{2}-80225137152y^{28}z^{4}+78103415097556992y^{26}z^{6}-9728886948999990755328y^{24}z^{8}-145356472724452054391586816y^{22}z^{10}+67543980904094501734596387274752y^{20}z^{12}-515279835690835231662740495287713792y^{18}z^{14}-169578715503140986218712366100851936198656y^{16}z^{16}-1167532625368065995483833564213510903487791104y^{14}z^{18}+179199029465372028667921798702232788793200358719488y^{12}z^{20}+5948582067223590007906211910485501845992316160421396480y^{10}z^{22}+87259962891744243433022392919157002554409065838019700850688y^{8}z^{24}+706508681625876583152737270393683364409886012125945555767525376y^{6}z^{26}+3246754316158672130522703230260508600169350023429790653263359508480y^{4}z^{28}+7875855484142385766938578576130513436047884577786465467315689038544896y^{2}z^{30}+7732351839687273514096325616431261048333858449822497115464138717036281856z^{32}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.ba.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.f.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.a.1 $48$ $2$ $2$ $1$ $1$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.96.1.d.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.w.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.bm.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.bt.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.dr.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.dt.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.ef.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.el.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.144.9.jc.1 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bfl.2 $48$ $4$ $4$ $9$ $2$ $1^{4}\cdot2^{2}$
240.96.1.nz.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.oh.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pf.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pn.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.sx.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tf.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ud.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ul.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.ew.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ikm.2 $240$ $6$ $6$ $17$ $?$ not computed