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 | $1^{2}\cdot2\cdot3^{2}\cdot4\cdot6\cdot8^{2}\cdot12\cdot24^{2}$ | 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: | 24X3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.6807 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&20\\0&5\end{bmatrix}$, $\begin{bmatrix}5&39\\24&35\end{bmatrix}$, $\begin{bmatrix}19&32\\24&43\end{bmatrix}$, $\begin{bmatrix}23&45\\0&1\end{bmatrix}$, $\begin{bmatrix}37&21\\12&13\end{bmatrix}$, $\begin{bmatrix}37&39\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.ge.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: | $2^{9}\cdot3^{3}$ |
| 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 |
|---|
| $(-1:1:0)$, $(0:0:1)$, $(-1/2:1/2:1)$, $(0:1:0)$, $(1:0:0)$, $(1:-1:1)$ |
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}+560x^{21}z^{3}-6228x^{20}z^{4}+21312x^{19}z^{5}+147196x^{18}z^{6}-1045656x^{17}z^{7}+1570404x^{16}z^{8}+16613880x^{15}z^{9}-47628300x^{14}z^{10}-50786592x^{13}z^{11}+322737334x^{12}z^{12}+276527760x^{11}z^{13}-1079319792x^{10}z^{14}-1863342528x^{9}z^{15}+850140072x^{8}z^{16}+6025195536x^{7}z^{17}+8968542576x^{6}z^{18}+6719770896x^{5}z^{19}-4624460340x^{4}z^{20}-49207331848x^{3}z^{21}+11811160042x^{2}y^{22}-894426932056x^{2}y^{21}z+18373869486924x^{2}y^{20}z^{2}-157833048494346x^{2}y^{19}z^{3}+658684976838788x^{2}y^{18}z^{4}-1324538283287116x^{2}y^{17}z^{5}+781842060423704x^{2}y^{16}z^{6}+1533180916135704x^{2}y^{15}z^{7}-2712934792475396x^{2}y^{14}z^{8}+590511563563876x^{2}y^{13}z^{9}+1595305833279964x^{2}y^{12}z^{10}-1031830379657292x^{2}y^{11}z^{11}-138135778525246x^{2}y^{10}z^{12}+173917644151932x^{2}y^{9}z^{13}-30921912488092x^{2}y^{8}z^{14}+98232060709206x^{2}y^{7}z^{15}-23486614976676x^{2}y^{6}z^{16}-32488352811808x^{2}y^{5}z^{17}-4933445877380x^{2}y^{4}z^{18}+4783644184738x^{2}y^{3}z^{19}+4312889936830x^{2}y^{2}z^{20}-737289766039x^{2}yz^{21}-191248575492x^{2}z^{22}+10737418220xy^{23}-651761280824xy^{22}z+10107131328280xy^{21}z^{2}-58882377815114xy^{20}z^{3}+130052506413260xy^{19}z^{4}-31075786477988xy^{18}z^{5}-113971840320512xy^{17}z^{6}-482813034156888xy^{16}z^{7}+1194272410814712xy^{15}z^{8}+56809270853940xy^{14}z^{9}-1887812686322648xy^{13}z^{10}+1063474622766836xy^{12}z^{11}+789876901492140xy^{11}z^{12}-749896014735380xy^{10}z^{13}-5157824564496xy^{9}z^{14}+37309477640126xy^{8}z^{15}+11514730665452xy^{7}z^{16}+79949572802544xy^{6}z^{17}-21062811739484xy^{5}z^{18}-18333386511586xy^{4}z^{19}-5534410996012xy^{3}z^{20}+2706524954719xy^{2}z^{21}+1898750455658xyz^{22}-492627769152xz^{23}+y^{24}+12y^{23}z+60y^{22}z^{2}+10737417660y^{21}z^{3}-673236123492y^{20}z^{4}+11432128705056y^{19}z^{5}-80379761791508y^{18}z^{6}+266781690274616y^{17}z^{7}-385644282930768y^{16}z^{8}+4980331768000y^{15}z^{9}+643272615743824y^{14}z^{10}-576713300840328y^{13}z^{11}-145576763014588y^{12}z^{12}+399238115623320y^{11}z^{13}-97776208396536y^{10}z^{14}-45115385998044y^{9}z^{15}+2774195711132y^{8}z^{16}-18510570043920y^{7}z^{17}+22552376006908y^{6}z^{18}+5411630274488y^{5}z^{19}-2301497655816y^{4}z^{20}-2982023291268y^{3}z^{21}-722246292710y^{2}z^{22}+492627769152yz^{23}+4096z^{24}}{z^{3}(x^{21}-9x^{20}z+30x^{19}z^{2}-69x^{18}z^{3}+186x^{17}z^{4}-276x^{16}z^{5}+325x^{15}z^{6}-1176x^{14}z^{7}-648x^{13}z^{8}-1292x^{12}z^{9}+8964x^{11}z^{10}+39012x^{10}z^{11}+95188x^{9}z^{12}+70851x^{8}z^{13}-494514x^{7}z^{14}-2699677x^{6}z^{15}-7357278x^{5}z^{16}-8617572x^{4}z^{17}+25797015x^{3}z^{18}+19x^{2}y^{19}-4776x^{2}y^{18}z+290428x^{2}y^{17}z^{2}-141249928x^{2}y^{16}z^{3}+5532970200x^{2}y^{15}z^{4}-60558790224x^{2}y^{14}z^{5}+269309883194x^{2}y^{13}z^{6}-518992842952x^{2}y^{12}z^{7}+256699185166x^{2}y^{11}z^{8}+519479629147x^{2}y^{10}z^{9}-707494546189x^{2}y^{9}z^{10}+21447871960x^{2}y^{8}z^{11}+407216749212x^{2}y^{7}z^{12}-168822360401x^{2}y^{6}z^{13}-81461650522x^{2}y^{5}z^{14}+64368033081x^{2}y^{4}z^{15}+550790371x^{2}y^{3}z^{16}-8131390728x^{2}y^{2}z^{17}+927957036x^{2}yz^{18}+185740116x^{2}z^{19}+17xy^{20}-4030xy^{19}z+210294xy^{18}z^{2}-121611283xy^{17}z^{3}+3508781048xy^{16}z^{4}-24831598368xy^{15}z^{5}+52488748093xy^{14}z^{6}+3316756218xy^{13}z^{7}-31736948060xy^{12}z^{8}-261108669365xy^{11}z^{9}+420570760763xy^{10}z^{10}+115089357544xy^{9}z^{11}-531655109686xy^{8}z^{12}+171025926854xy^{7}z^{13}+219830270756xy^{6}z^{14}-143018891300xy^{5}z^{15}-23864765547xy^{4}z^{16}+37321461668xy^{3}z^{17}-4074392653xy^{2}z^{18}-2889332952xyz^{19}+583409688xz^{20}-y^{21}-9y^{20}z-30y^{19}z^{2}-52y^{18}z^{3}-4250y^{17}z^{4}+218112y^{16}z^{5}-122040224y^{15}z^{6}+3752415414y^{14}z^{7}-32091971436y^{13}z^{8}+108942235548y^{12}z^{9}-144208587002y^{11}z^{10}-9064300608y^{10}z^{11}+193639231414y^{9}z^{12}-123185955500y^{8}z^{13}-61844349864y^{7}z^{14}+84093173382y^{6}z^{15}-6347704678y^{5}z^{16}-20238105384y^{4}z^{17}+5822705290y^{3}z^{18}+1536773460y^{2}z^{19}-583409688yz^{20})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.1-24.ir.1.16 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.fz.1.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.ga.1.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gb.3.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gc.2.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gh.2.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gi.3.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gj.3.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gk.2.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-24.db.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ds.1.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.dy.3.21 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ed.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.eg.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ek.1.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.ep.4.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.es.3.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.eu.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ev.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ew.3.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ex.3.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ex.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ez.3.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fb.3.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fc.3.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fd.3.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fd.3.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fe.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ff.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ff.1.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fh.1.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fj.3.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fl.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gl.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gn.3.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hf.4.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hh.3.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hj.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hl.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.id.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.if.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.ban.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bap.3.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbh.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbj.3.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfd.3.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bff.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfx.3.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfz.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhh.1.19 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhj.1.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhl.3.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhn.3.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhp.3.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhr.3.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bht.3.21 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhv.3.19 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-24.lp.2.18 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-120.bep.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.beq.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.ber.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bes.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bff.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfg.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfh.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfi.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-120.le.1.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lk.1.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lq.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lw.2.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mi.1.19 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mo.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mu.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.na.3.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ng.1.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nh.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ni.4.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nj.3.12 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nw.3.12 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nx.4.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ny.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nz.1.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ut.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uv.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ux.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uz.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vj.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vl.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vn.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vp.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xv.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xx.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zf.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zh.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zr.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zt.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbb.3.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbd.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fsv.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsx.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuf.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuh.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fur.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fut.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwb.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwd.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdh.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdj.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdl.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdn.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdx.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdz.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.geb.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ged.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |