Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $72$ | ||
| 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 | $2^{2}\cdot4^{3}\cdot6^{2}\cdot8\cdot12^{3}\cdot24$ | 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: | 24AA3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.1583 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\0&17\end{bmatrix}$, $\begin{bmatrix}7&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&10\\0&23\end{bmatrix}$, $\begin{bmatrix}11&14\\0&1\end{bmatrix}$, $\begin{bmatrix}17&2\\0&11\end{bmatrix}$, $\begin{bmatrix}23&2\\12&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^5$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.bp.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{9}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 72.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y - 2 x^{3} z + 3 x^{2} y^{2} - 3 x^{2} y z + 2 x y^{3} - 3 x y z^{2} + 2 x z^{3} + 3 y^{4} + \cdots + y z^{3} $ |
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 |
|---|
| $(1:0:0)$, $(1:0:1)$, $(-1:0:1)$, $(0:0:1)$ |
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{1}{3^2}\cdot\frac{2985984x^{24}+71663616x^{23}z+716636160x^{22}z^{2}+4323704832x^{21}z^{3}+15604752384x^{20}z^{4}+46509686784x^{19}z^{5}-29047652352x^{18}z^{6}+1110857711616x^{17}z^{7}-14041079156736x^{16}z^{8}+183397465128960x^{15}z^{9}-2441228245401600x^{14}z^{10}+32857563121385472x^{13}z^{11}-446654764769624064x^{12}z^{12}+6123609159470088192x^{11}z^{13}-84577133074172608512x^{10}z^{14}+1175735835911244939264x^{9}z^{15}-16437967588718151634944x^{8}z^{16}+230989624927861298135040x^{7}z^{17}-3260695719457365694414848x^{6}z^{18}+46217194328068489744515072x^{5}z^{19}-657510307178647411266379776x^{4}z^{20}+9385566494428784663570644992x^{3}z^{21}-6835089540240428x^{2}y^{22}-58086185351755040x^{2}y^{21}z-510542285879954976x^{2}y^{20}z^{2}-3333360580559836544x^{2}y^{19}z^{3}-19874339274351685184x^{2}y^{18}z^{4}-106148344345564773888x^{2}y^{17}z^{5}-525710122380521493504x^{2}y^{16}z^{6}-2433419300850046457856x^{2}y^{15}z^{7}-10636082948434543992576x^{2}y^{14}z^{8}-44161975092419645767168x^{2}y^{13}z^{9}-174997399663220879848960x^{2}y^{12}z^{10}-663835267166243569858560x^{2}y^{11}z^{11}-2414957375452028933776384x^{2}y^{10}z^{12}-8428945287527990490099712x^{2}y^{9}z^{13}-28194230466698764047593472x^{2}y^{8}z^{14}-90079357961283992319787008x^{2}y^{7}z^{15}-273005536571529063827030016x^{2}y^{6}z^{16}-774325307632104147026313216x^{2}y^{5}z^{17}-1999884489740479642680885248x^{2}y^{4}z^{18}-4400021762863429828836196352x^{2}y^{3}z^{19}-6093271950532374277292752896x^{2}y^{2}z^{20}+13530625162779787123114639360x^{2}yz^{21}+660787525891936616198176768x^{2}z^{22}-2243960695629084xy^{23}-25016440727687848xy^{22}z-219963581958968776xy^{21}z^{2}-1559466613548237264xy^{20}z^{3}-9615406645090592416xy^{19}z^{4}-53513649690452808256xy^{18}z^{5}-273586900323566813184xy^{17}z^{6}-1305074731721840500224xy^{16}z^{7}-5861908318341212435712xy^{15}z^{8}-24972823441877322137856xy^{14}z^{9}-101415760622105167845632xy^{13}z^{10}-394028500147681166540288xy^{12}z^{11}-1468062614072976838588416xy^{11}z^{12}-5250993534860296698816512xy^{10}z^{13}-18027040141726425469349888xy^{9}z^{14}-59285051324473873497145344xy^{8}z^{15}-185899215798226504815919104xy^{7}z^{16}-550911374183341339326971904xy^{6}z^{17}-1522117779441979072750583808xy^{5}z^{18}-3805485310250273378341027840xy^{4}z^{19}-6964506876232529216082018304xy^{3}z^{20}+6288997797304886919783383040xy^{2}z^{21}+8814617232642653021521313792xyz^{22}-9432015860274268207773024256xz^{23}-7674989513889519y^{24}-71896740394350252y^{23}z-633776176179267068y^{22}z^{2}-4277826888860914184y^{21}z^{3}-25892227687452826176y^{20}z^{4}-140775938071012919456y^{19}z^{5}-707159789041043480960y^{18}z^{6}-3317610421198842236928y^{17}z^{7}-14679977993053337753088y^{16}z^{8}-61665422781117400714752y^{15}z^{9}-247094821101781250808576y^{14}z^{10}-947603165677960189638400y^{13}z^{11}-3485074950214089651741184y^{12}z^{12}-12301274991444696535984128y^{11}z^{13}-41642418506654793568267264y^{10}z^{14}-134837094962257989201805312y^{9}z^{15}-415232286438249885568720896y^{8}z^{16}-1203067603945095108526620672y^{7}z^{17}-3210335816052721348945231872y^{6}z^{18}-7446547110371424912742711296y^{5}z^{19}-11763208087167162072066031616y^{4}z^{20}+3513404752221281488135389184y^{3}z^{21}-2831900160357358799118336000y^{2}z^{22}-4716007930137134103886561280yz^{23}+4096z^{24}}{y^{4}(y-2z)^{2}(y^{2}-yz+z^{2})^{2}(18656x^{2}y^{12}-59700x^{2}y^{11}z+145851x^{2}y^{10}z^{2}-229900x^{2}y^{9}z^{3}+282861x^{2}y^{8}z^{4}-273834x^{2}y^{7}z^{5}+228486x^{2}y^{6}z^{6}-152496x^{2}y^{5}z^{7}+85356x^{2}y^{4}z^{8}-37120x^{2}y^{3}z^{9}+13056x^{2}y^{2}z^{10}-3072x^{2}yz^{11}+512x^{2}z^{12}+5485xy^{13}-799xy^{12}z+9540xy^{11}z^{2}+19345xy^{10}z^{3}-70312xy^{9}z^{4}+144729xy^{8}z^{5}-166395xy^{7}z^{6}+155262xy^{6}z^{7}-110628xy^{5}z^{8}+66976xy^{4}z^{9}-30592xy^{3}z^{10}+11520xy^{2}z^{11}-2816xyz^{12}+512xz^{13}+20832y^{14}-43952y^{13}z+103953y^{12}z^{2}-141741y^{11}z^{3}+155860y^{10}z^{4}-110817y^{9}z^{5}+61839y^{8}z^{6}-13737y^{7}z^{7}-6426y^{6}z^{8}+13308y^{5}z^{9}-8444y^{4}z^{10}+4224y^{3}z^{11}-1152y^{2}z^{12}+256yz^{13})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.b.1.9 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.48.0-24.i.2.2 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-12.b.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.cd.3.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.cd.4.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ch.1.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ch.3.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.cl.3.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.cl.4.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.cp.1.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.cp.3.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.7-24.f.1.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.h.1.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.m.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.o.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.v.2.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.y.2.3 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.ba.1.2 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 24.384.7-24.bd.2.2 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 24.384.7-24.bi.3.14 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bi.4.11 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bm.1.7 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bm.3.5 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bq.3.11 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bq.4.5 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bu.1.5 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.bu.3.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.by.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.by.2.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.cc.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.cc.2.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.cg.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.cg.2.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.ck.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.ck.2.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 24.384.7-24.cp.2.8 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 24.384.7-24.cs.2.16 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 24.384.7-24.cu.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.cx.1.2 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.df.1.8 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.dh.1.16 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.dm.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.7-24.do.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.br.2.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 24.384.9-24.bx.1.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 24.384.9-24.cj.1.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.cl.1.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.dr.2.8 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.ea.1.20 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.ef.2.4 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 24.384.9-24.ei.2.4 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 24.384.9-24.fc.1.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fc.2.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fg.1.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fg.2.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fk.3.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fk.4.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fo.1.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.fo.2.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.576.13-24.bo.1.12 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 120.384.5-120.nd.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.nd.4.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.nh.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.nh.4.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.nl.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.nl.3.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.np.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.np.3.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.7-120.cj.2.17 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.cl.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.cr.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ct.2.21 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.di.2.37 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.dk.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.dq.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ds.2.43 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ei.2.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ei.3.26 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.em.2.22 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.em.3.30 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.eq.2.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.eq.4.21 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.eu.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.eu.4.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fo.3.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fo.4.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fs.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fs.3.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fw.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.fw.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ga.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ga.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.gs.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.gu.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ha.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.hc.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.hq.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.hs.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.hy.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.7-120.ia.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.384.9-120.ji.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.jk.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.jq.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.js.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.kg.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ki.2.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ko.2.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.kq.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.pg.3.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.pg.4.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.pk.2.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.pk.3.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.po.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.po.2.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ps.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ps.2.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.5-168.nd.1.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.nd.4.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.nh.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
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| 264.384.9-264.po.4.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
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| 312.384.9-312.jq.1.58 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
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| 312.384.9-312.pk.3.47 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
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| 312.384.9-312.po.2.31 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
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