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Properties

Label 24.96.1.do.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $576$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $2$ are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2253

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&21\\12&5\end{bmatrix}$, $\begin{bmatrix}7&12\\20&23\end{bmatrix}$, $\begin{bmatrix}11&3\\0&5\end{bmatrix}$, $\begin{bmatrix}13&9\\16&23\end{bmatrix}$, $\begin{bmatrix}17&3\\4&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035912
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.do.2.1, 24.192.1-24.do.2.2, 24.192.1-24.do.2.3, 24.192.1-24.do.2.4, 24.192.1-24.do.2.5, 24.192.1-24.do.2.6, 24.192.1-24.do.2.7, 24.192.1-24.do.2.8, 24.192.1-24.do.2.9, 24.192.1-24.do.2.10, 24.192.1-24.do.2.11, 24.192.1-24.do.2.12, 24.192.1-24.do.2.13, 24.192.1-24.do.2.14, 24.192.1-24.do.2.15, 24.192.1-24.do.2.16, 120.192.1-24.do.2.1, 120.192.1-24.do.2.2, 120.192.1-24.do.2.3, 120.192.1-24.do.2.4, 120.192.1-24.do.2.5, 120.192.1-24.do.2.6, 120.192.1-24.do.2.7, 120.192.1-24.do.2.8, 120.192.1-24.do.2.9, 120.192.1-24.do.2.10, 120.192.1-24.do.2.11, 120.192.1-24.do.2.12, 120.192.1-24.do.2.13, 120.192.1-24.do.2.14, 120.192.1-24.do.2.15, 120.192.1-24.do.2.16, 168.192.1-24.do.2.1, 168.192.1-24.do.2.2, 168.192.1-24.do.2.3, 168.192.1-24.do.2.4, 168.192.1-24.do.2.5, 168.192.1-24.do.2.6, 168.192.1-24.do.2.7, 168.192.1-24.do.2.8, 168.192.1-24.do.2.9, 168.192.1-24.do.2.10, 168.192.1-24.do.2.11, 168.192.1-24.do.2.12, 168.192.1-24.do.2.13, 168.192.1-24.do.2.14, 168.192.1-24.do.2.15, 168.192.1-24.do.2.16, 264.192.1-24.do.2.1, 264.192.1-24.do.2.2, 264.192.1-24.do.2.3, 264.192.1-24.do.2.4, 264.192.1-24.do.2.5, 264.192.1-24.do.2.6, 264.192.1-24.do.2.7, 264.192.1-24.do.2.8, 264.192.1-24.do.2.9, 264.192.1-24.do.2.10, 264.192.1-24.do.2.11, 264.192.1-24.do.2.12, 264.192.1-24.do.2.13, 264.192.1-24.do.2.14, 264.192.1-24.do.2.15, 264.192.1-24.do.2.16, 312.192.1-24.do.2.1, 312.192.1-24.do.2.2, 312.192.1-24.do.2.3, 312.192.1-24.do.2.4, 312.192.1-24.do.2.5, 312.192.1-24.do.2.6, 312.192.1-24.do.2.7, 312.192.1-24.do.2.8, 312.192.1-24.do.2.9, 312.192.1-24.do.2.10, 312.192.1-24.do.2.11, 312.192.1-24.do.2.12, 312.192.1-24.do.2.13, 312.192.1-24.do.2.14, 312.192.1-24.do.2.15, 312.192.1-24.do.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 24x - 56 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2\cdot3}\cdot\frac{4368x^{2}y^{30}+645301842048x^{2}y^{28}z^{2}+70151098400547840x^{2}y^{26}z^{4}+186522479485271875584x^{2}y^{24}z^{6}+119157968987634843058176x^{2}y^{22}z^{8}+32720491871504697018286080x^{2}y^{20}z^{10}+4231346102449358655505563648x^{2}y^{18}z^{12}+252231107820501546717683908608x^{2}y^{16}z^{14}+16348952592469379622532590600192x^{2}y^{14}z^{16}+3859374595689513044015958737289216x^{2}y^{12}z^{18}+508276307301266905942089473124728832x^{2}y^{10}z^{20}+16477875190992751824359614340250206208x^{2}y^{8}z^{22}-2811941752790880495194283284662503604224x^{2}y^{6}z^{24}-346979773530222101129979284800266974527488x^{2}y^{4}z^{26}-15660813860898570166612862160809325194379264x^{2}y^{2}z^{28}-241624017713473993193847791559410735778889728x^{2}z^{30}+6644256xy^{30}z+51447878942976xy^{28}z^{3}+1351621853937027072xy^{26}z^{5}+1527719524413846994944xy^{24}z^{7}+699754572703515829469184xy^{22}z^{9}+166314658248090415600238592xy^{20}z^{11}+21970648158878567680306053120xy^{18}z^{13}+1235644579575780462096298278912xy^{16}z^{15}-96088325125661482697955068608512xy^{14}z^{17}-19840906163434867375893334865215488xy^{12}z^{19}+104076930420332462054622952729608192xy^{10}z^{21}+291473567865387044334546359239078576128xy^{8}z^{23}+31491217317228151604332026691769270796288xy^{6}z^{25}+1387934703290791434192813218350159361998848xy^{4}z^{27}+15660820305528097277643181074731036653387776xy^{2}z^{29}-483248035426947986387695583118821471557779456xz^{31}+y^{32}+3920240640y^{30}z^{2}+2393437997357568y^{28}z^{4}+19159728774688161792y^{26}z^{6}+16972237199959727063040y^{24}z^{8}+6474209606200560193634304y^{22}z^{10}+1370813139566585215526633472y^{20}z^{12}+182283963970642279038965514240y^{18}z^{14}+17233774450660421922802353831936y^{16}z^{16}+1140153460968270812613991578206208y^{14}z^{18}+10480792157288886595308171445665792y^{12}z^{20}-8579197818383539731290031525910806528y^{10}z^{22}-1007830455441222537830019423999363121152y^{8}z^{24}-46510179330538580065154040120607852462080y^{6}z^{26}-82788656682241907748641006039432958050304y^{4}z^{28}+71592361787004743642503545387404220941991936y^{2}z^{30}+1932992164160049652905339572562497010023792640z^{32}}{y^{2}(y^{2}+216z^{2})(x^{2}y^{26}+9288x^{2}y^{24}z^{2}-47402496x^{2}y^{22}z^{4}-1750939213824x^{2}y^{20}z^{6}-11196286475563008x^{2}y^{18}z^{8}-12381744651832885248x^{2}y^{16}z^{10}-2329743969511142326272x^{2}y^{14}z^{12}+2973240048656863538970624x^{2}y^{12}z^{14}+1760198220255865738295771136x^{2}y^{10}z^{16}+357357673322335980452396924928x^{2}y^{8}z^{18}+28549577959486109664253958946816x^{2}y^{6}z^{20}+685187621379459553759445026603008x^{2}y^{4}z^{22}-51572123992452677730859745280x^{2}y^{2}z^{24}+2227915756473955677973140996096x^{2}z^{26}-100xy^{26}z-3581712xy^{24}z^{3}-42249807360xy^{22}z^{5}-197291498010624xy^{20}z^{7}-368962855819001856xy^{18}z^{9}-407376509217946140672xy^{16}z^{11}-271815531140750990376960xy^{14}z^{13}-102824402379830472311046144xy^{12}z^{15}-20276343573955968568884461568xy^{10}z^{17}-1781897594342764726671284109312xy^{8}z^{19}-38065934750741586617742032633856xy^{6}z^{21}+1370373523688119359096299024547840xy^{4}z^{23}-41257699193962142184687796224xy^{2}z^{25}+4455831512947911355946281992192xz^{27}+3292y^{26}z^{2}+89761824y^{24}z^{4}+834238020096y^{22}z^{6}+3279513581678592y^{20}z^{8}+5052419953109581824y^{18}z^{10}+3969554533486396047360y^{16}z^{12}+1679661467181985264828416y^{14}z^{14}+365005160132446939112275968y^{12}z^{16}+33511649652690264669435199488y^{10}z^{18}+19562868001740981629444161536y^{8}z^{20}-152262676079560329184983348412416y^{6}z^{22}-5481538790593270895372562843303936y^{4}z^{24}-1196473276624902123355946090496y^{2}z^{26}+62381641181270758983247947890688z^{28})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0.bt.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iv.1 $24$ $2$ $2$ $1$ $0$ 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
24.192.5.cv.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dn.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ej.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.em.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.et.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fa.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ft.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fx.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.s.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.y.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ez.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fp.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bam.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bao.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbe.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcy.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bda.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdq.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bam.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bao.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbe.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcy.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bda.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdo.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdq.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bam.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bao.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbe.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcy.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bda.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdo.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdq.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bam.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bao.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbc.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbe.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcy.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bda.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdo.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdq.3 $312$ $2$ $2$ $5$ $?$ not computed