↑

Properties

Label 48.48.1.gc.1
Level $48$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $6$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $4^{4}\cdot16^{2}$ Cusp orbits $2\cdot4$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-12$)

Other labels

Cummins and Pauli (CP) label: 16H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.142

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&18\\42&5\end{bmatrix}$, $\begin{bmatrix}13&3\\12&23\end{bmatrix}$, $\begin{bmatrix}37&39\\2&35\end{bmatrix}$, $\begin{bmatrix}43&19\\6&5\end{bmatrix}$, $\begin{bmatrix}47&33\\44&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 48-isogeny field degree: $32$
Cyclic 48-torsion field degree: $512$
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} + 9x $
 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 -2^4\cdot3^3\,\frac{175560x^{2}y^{30}+500727360x^{2}y^{29}z+312595245588x^{2}y^{28}z^{2}+70560223799616x^{2}y^{27}z^{3}+6182205724838016x^{2}y^{26}z^{4}+138932689973285520x^{2}y^{25}z^{5}-4075444306339406586x^{2}y^{24}z^{6}-21175356458392951104x^{2}y^{23}z^{7}+2973542273035837587504x^{2}y^{22}z^{8}-65324181571305497701056x^{2}y^{21}z^{9}+428570092671140717322219x^{2}y^{20}z^{10}+4027200498825847022775312x^{2}y^{19}z^{11}-393156115632738232670429928x^{2}y^{18}z^{12}+2848630236959660778923504616x^{2}y^{17}z^{13}-135855184731729188105802939543x^{2}y^{16}z^{14}+541716891044906978541141117120x^{2}y^{15}z^{15}-6829728356882752147477413107784x^{2}y^{14}z^{16}+396245781429774584508722953930752x^{2}y^{13}z^{17}+1704253332955837004202039781066632x^{2}y^{12}z^{18}+30683334033601036996246592301491328x^{2}y^{11}z^{19}-319163627939027538043406772169108128x^{2}y^{10}z^{20}+484810203581348123152017784209913992x^{2}y^{9}z^{21}-17809573614229339860949030623191302809x^{2}y^{8}z^{22}-25972560965099623750095275691594767712x^{2}y^{7}z^{23}-63100973492047841739270991913406765960x^{2}y^{6}z^{24}-211315420181944622316544866162329759136x^{2}y^{5}z^{25}+5326509830581419322472919952794141983361x^{2}y^{4}z^{26}-2081218804406922361921295026941838462608x^{2}y^{3}z^{27}+42340364330015884933275101806520554373160x^{2}y^{2}z^{28}-98792783733390393227951527668287515013400x^{2}yz^{29}+7885204634687427929086217123476949245125x^{2}z^{30}+6600xy^{31}+43947384xy^{30}z+41431536336xy^{29}z^{2}+12829902248448xy^{28}z^{3}+1552005804237696xy^{27}z^{4}+61473348214254378xy^{26}z^{5}-606455542636132896xy^{25}z^{6}-33681319360371202464xy^{24}z^{7}+1132349448594594324528xy^{23}z^{8}-8730191231706951579816xy^{22}z^{9}-210099186721351077151392xy^{21}z^{10}+10534063938292837575710712xy^{20}z^{11}-126970415101289853622181736xy^{19}z^{12}+2985259221238762428662956293xy^{18}z^{13}-14279119007236186094744350752xy^{17}z^{14}-115881190168327171268386262688xy^{16}z^{15}-8995811371000643429674832609352xy^{15}z^{16}-115418556040637412704758927236192xy^{14}z^{17}-386651023771193561646269799885648xy^{13}z^{18}+3506030395189435841013109382082336xy^{12}z^{19}+100412838576639468804913343966051424xy^{11}z^{20}+1676118956303654114015752470128834325xy^{10}z^{21}-1413523874437746540848427290372433360xy^{9}z^{22}+47025379922617165379599869892153544592xy^{8}z^{23}-302133665849997355480396359392311434888xy^{7}z^{24}+127154771690710347998846155322806469652xy^{6}z^{25}-6039699523275957559178156126055845502480xy^{5}z^{26}+3688771313530175307140845251473418647784xy^{4}z^{27}-12912272709500306184984710074939925610072xy^{3}z^{28}+101523398188078951839302120051109125926365xy^{2}z^{29}+121868791845274463870244625406738458214400xyz^{30}-213816636124234384863640344028703096832000xz^{31}+125y^{32}+3168752y^{31}z+4872931264y^{30}z^{2}+2112843515328y^{29}z^{3}+350723950578072y^{28}z^{4}+21650669647082496y^{27}z^{5}+166055088116664720y^{26}z^{6}-14383110424092715152y^{25}z^{7}+190179032617385479116y^{24}z^{8}+3664656895895389197216y^{23}z^{9}-178684733316204834512832y^{22}z^{10}+3088194446322392297846328y^{21}z^{11}-37351547329428922241670390y^{20}z^{12}+66701668683262036086729360y^{19}z^{13}+7455075070784471824632964200y^{18}z^{14}+66600465313077166740192498312y^{17}z^{15}+4973995737551597366763439932636y^{16}z^{16}-2122641652075956333644807795184y^{15}z^{17}-48028508858618252316295071795072y^{14}z^{18}-9297962623648189619702251023871584y^{13}z^{19}+3873123477798407345779775793902448y^{12}z^{20}-450679842776012683002771092707770048y^{11}z^{21}-528728624040203731262263718255511672y^{10}z^{22}+20041550530494351353523084125588131800y^{9}z^{23}-28774696816374261320123855419604933034y^{8}z^{24}+635552444371702791996255007202197546512y^{7}z^{25}-832656901362965969404430929642388479008y^{6}z^{26}+2489404370046571636882893729903517119432y^{5}z^{27}-11224622577879426665055765613514652425538y^{4}z^{28}-13644263269917573342503802423191658562320y^{3}z^{29}+23805414722280396112973437720057125028200y^{2}z^{30}-13784724611682865073060235178630167000yz^{31}+1845891563353309582719917015994598125z^{32}}{192x^{2}y^{30}-6294996x^{2}y^{28}z^{2}-121163579136x^{2}y^{26}z^{4}-148739031441342x^{2}y^{24}z^{6}+5693267998883366784x^{2}y^{22}z^{8}+29209296254438788734681x^{2}y^{20}z^{10}+50640030791263518404547456x^{2}y^{18}z^{12}+36546904108227039184093972563x^{2}y^{16}z^{14}+11670789785770231579624406501952x^{2}y^{14}z^{16}+1024664989338674554689358745023368x^{2}y^{12}z^{18}+45422821958244260795554594646179584x^{2}y^{10}z^{20}+951552448451617707496002650806697805x^{2}y^{8}z^{22}+4814985757523919559893769414521958848x^{2}y^{6}z^{24}-113482661198955536990864723883346477413x^{2}y^{4}z^{26}-1419882506645177658004461527050654837632x^{2}y^{2}z^{28}-3373248250320423699383139089387777002713x^{2}z^{30}-15192xy^{30}z-360013248xy^{28}z^{3}-5882175123570xy^{26}z^{5}-53521973835931008xy^{24}z^{7}-224261347367199391992xy^{22}z^{9}-407746494783578444721984xy^{20}z^{11}-310542415417204799996852217xy^{18}z^{13}-109456791020307826859852774784xy^{16}z^{15}-29714979785558678506378144667232xy^{14}z^{17}-1483509748189805435247259672054080xy^{12}z^{19}+9637008675600306127913256438585927xy^{10}z^{21}+2633499362027050772349135157195394880xy^{8}z^{23}+72855704140949776074972587791392162492xy^{6}z^{25}+717562567863325130350929238830227624064xy^{4}z^{27}+1488553360301829251439400855470852294063xy^{2}z^{29}-6452057374420298495433759069822309629952xz^{31}-y^{32}+582336y^{30}z^{2}+15064201800y^{28}z^{4}+159101104079232y^{26}z^{6}+711418656432270564y^{24}z^{8}+1047380932743477033600y^{22}z^{10}-503773662166825355370450y^{20}z^{12}-1925790180807215979796937472y^{18}z^{14}-998516958797165649360553148796y^{16}z^{16}-72843830424020239412730667441344y^{14}z^{18}-5942814683910119620227542678457456y^{12}z^{20}-286732134239955578077789302593020992y^{10}z^{22}-6581431354354086481260266358465650622y^{8}z^{24}-63183155765547416379744649780283806656y^{6}z^{26}-123749777874604249499748455955631978134y^{4}z^{28}+716895263936741493452682688306468403328y^{2}z^{30}-31158766826903324165375390625z^{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.bq.1 $8$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.96.3.o.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.en.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.iw.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.jh.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.tp.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.ua.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.uz.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.vk.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.xj.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.xk.1 $48$ $2$ $2$ $3$ $3$ $1^{2}$
48.96.3.xr.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.xs.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.zl.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.zm.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.zp.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.zq.1 $48$ $2$ $2$ $3$ $3$ $1^{2}$
48.96.5.lr.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.ls.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.96.5.lv.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.lw.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.96.5.qn.1 $48$ $2$ $2$ $5$ $5$ $1^{2}\cdot2$
48.96.5.qo.1 $48$ $2$ $2$ $5$ $4$ $1^{2}\cdot2$
48.96.5.qv.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.qw.1 $48$ $2$ $2$ $5$ $4$ $1^{2}\cdot2$
48.144.7.sd.1 $48$ $3$ $3$ $7$ $3$ $1^{6}$
48.192.11.ln.1 $48$ $4$ $4$ $11$ $4$ $1^{10}$
240.96.3.ezn.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ezt.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fax.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fbd.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fie.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fiq.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fky.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.flk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fpp.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fpq.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fqf.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fqg.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ftv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ftw.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fud.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fue.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.dih.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dii.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dip.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.diq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dmr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dms.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dnh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dni.1 $240$ $2$ $2$ $5$ $?$ not computed
240.240.17.bak.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.xea.1 $240$ $6$ $6$ $17$ $?$ not computed