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Properties

Label 24.96.1.da.3
Level $24$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $12$ 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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1637

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&0\\0&5\end{bmatrix}$, $\begin{bmatrix}17&15\\8&23\end{bmatrix}$, $\begin{bmatrix}19&15\\16&17\end{bmatrix}$, $\begin{bmatrix}23&3\\12&23\end{bmatrix}$, $\begin{bmatrix}23&15\\8&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035912
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.da.3.1, 24.192.1-24.da.3.2, 24.192.1-24.da.3.3, 24.192.1-24.da.3.4, 24.192.1-24.da.3.5, 24.192.1-24.da.3.6, 24.192.1-24.da.3.7, 24.192.1-24.da.3.8, 24.192.1-24.da.3.9, 24.192.1-24.da.3.10, 24.192.1-24.da.3.11, 24.192.1-24.da.3.12, 24.192.1-24.da.3.13, 24.192.1-24.da.3.14, 24.192.1-24.da.3.15, 24.192.1-24.da.3.16, 120.192.1-24.da.3.1, 120.192.1-24.da.3.2, 120.192.1-24.da.3.3, 120.192.1-24.da.3.4, 120.192.1-24.da.3.5, 120.192.1-24.da.3.6, 120.192.1-24.da.3.7, 120.192.1-24.da.3.8, 120.192.1-24.da.3.9, 120.192.1-24.da.3.10, 120.192.1-24.da.3.11, 120.192.1-24.da.3.12, 120.192.1-24.da.3.13, 120.192.1-24.da.3.14, 120.192.1-24.da.3.15, 120.192.1-24.da.3.16, 168.192.1-24.da.3.1, 168.192.1-24.da.3.2, 168.192.1-24.da.3.3, 168.192.1-24.da.3.4, 168.192.1-24.da.3.5, 168.192.1-24.da.3.6, 168.192.1-24.da.3.7, 168.192.1-24.da.3.8, 168.192.1-24.da.3.9, 168.192.1-24.da.3.10, 168.192.1-24.da.3.11, 168.192.1-24.da.3.12, 168.192.1-24.da.3.13, 168.192.1-24.da.3.14, 168.192.1-24.da.3.15, 168.192.1-24.da.3.16, 264.192.1-24.da.3.1, 264.192.1-24.da.3.2, 264.192.1-24.da.3.3, 264.192.1-24.da.3.4, 264.192.1-24.da.3.5, 264.192.1-24.da.3.6, 264.192.1-24.da.3.7, 264.192.1-24.da.3.8, 264.192.1-24.da.3.9, 264.192.1-24.da.3.10, 264.192.1-24.da.3.11, 264.192.1-24.da.3.12, 264.192.1-24.da.3.13, 264.192.1-24.da.3.14, 264.192.1-24.da.3.15, 264.192.1-24.da.3.16, 312.192.1-24.da.3.1, 312.192.1-24.da.3.2, 312.192.1-24.da.3.3, 312.192.1-24.da.3.4, 312.192.1-24.da.3.5, 312.192.1-24.da.3.6, 312.192.1-24.da.3.7, 312.192.1-24.da.3.8, 312.192.1-24.da.3.9, 312.192.1-24.da.3.10, 312.192.1-24.da.3.11, 312.192.1-24.da.3.12, 312.192.1-24.da.3.13, 312.192.1-24.da.3.14, 312.192.1-24.da.3.15, 312.192.1-24.da.3.16
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.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 24x + 56 $
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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 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2\cdot3}\cdot\frac{4272x^{2}y^{30}+381796219008x^{2}y^{28}z^{2}+6812128491678720x^{2}y^{26}z^{4}-10373657205488541696x^{2}y^{24}z^{6}+6986428868461841547264x^{2}y^{22}z^{8}-2634370042417432292229120x^{2}y^{20}z^{10}+646898812648594024559542272x^{2}y^{18}z^{12}-137695105189142014154005020672x^{2}y^{16}z^{14}+22486801364609119137280146014208x^{2}y^{14}z^{16}-3121909200150020884737343560351744x^{2}y^{12}z^{18}+421648702271548555769774695941931008x^{2}y^{10}z^{20}-42878097963421470533574818876601925632x^{2}y^{8}z^{22}+4092780413133716358824252244064116670464x^{2}y^{6}z^{24}-346980158514064819829520438559031099916288x^{2}y^{4}z^{26}+15660818018724071528567906621403977748578304x^{2}y^{2}z^{28}-241624017713473993193847791559410735778889728x^{2}z^{30}+6264096xy^{30}z+18756378434304xy^{28}z^{3}+32534183808755712xy^{26}z^{5}-69857255703050993664xy^{24}z^{7}+47561467484101457608704xy^{22}z^{9}-18152206192051411492012032xy^{20}z^{11}+5399805297836754395253964800xy^{18}z^{13}-1218880385033744283960430559232xy^{16}z^{15}+207403431097121891311614393581568xy^{14}z^{17}-31742510125663923215959913285025792xy^{12}z^{19}+3721682007161439395621472210813714432xy^{10}z^{21}-350303856870675381937077436594098536448xy^{8}z^{23}+28929529837246630355834197215443102466048xy^{6}z^{25}-1387933817827953181183868564705001873604608xy^{4}z^{27}+15660811989877094553733092153541731544989696xy^{2}z^{29}+483248035426947986387695583118821471557779456xz^{31}+y^{32}+3403956480y^{30}z^{2}+488847078885888y^{28}z^{4}-421493414224822272y^{26}z^{6}+114604470433959690240y^{24}z^{8}+11246842580227631087616y^{22}z^{10}-27829063978103533845086208y^{20}z^{12}+12144649626854281062855475200y^{18}z^{14}-2784465286388962786555560198144y^{16}z^{16}+533309682535007643570745777324032y^{14}z^{18}-75180389137889052879071568910614528y^{12}z^{20}+7775547706418979236884542784586907648y^{10}z^{22}-680902790119066893638923189658490765312y^{8}z^{24}+36339561034909681012924603438016652902400y^{6}z^{26}-82799898210449293775242695795345419403264y^{4}z^{28}-71592245367890705507762300490753949424418816y^{2}z^{30}+1932992164160049652905339572562497010023792640z^{32}}{y^{2}(y^{2}-216z^{2})^{2}(x^{2}y^{24}-603504x^{2}y^{22}z^{2}-5436077184x^{2}y^{20}z^{4}-23362538130432x^{2}y^{18}z^{6}-11881402594430976x^{2}y^{16}z^{8}+42201914071903764480x^{2}y^{14}z^{10}-10518679678018793766912x^{2}y^{12}z^{12}-2450863252611973628559360x^{2}y^{10}z^{14}+1110711455450855709201137664x^{2}y^{8}z^{16}-117495353597709978362015907840x^{2}y^{6}z^{18}+3172336025007658077986815475712x^{2}y^{4}z^{20}-2005582599706493022866767872x^{2}y^{2}z^{22}+10314424798490535546171949056x^{2}z^{24}-188xy^{24}z+11418192xy^{22}z^{3}+75065678208xy^{20}z^{5}+10436965862400xy^{18}z^{7}-470323954713894912xy^{16}z^{9}+993893947056193536xy^{14}z^{11}+171893901913005996638208xy^{12}z^{13}-61195606635407507277742080xy^{10}z^{15}+7297069050708177794321350656xy^{8}z^{17}-205583951810487377106258886656xy^{6}z^{19}-6344726434199567456379674296320xy^{4}z^{21}+4297676999371056477571645440xy^{2}z^{23}-20628849596981071092343898112xz^{25}+14812y^{24}z^{2}+16937856y^{22}z^{4}+618507394560y^{20}z^{6}+3557875105161216y^{18}z^{8}+885146228992917504y^{16}z^{10}-2867060551868470001664y^{14}z^{12}+1115139167572661253439488y^{12}z^{14}-146710750686379534706540544y^{10}z^{16}+2816713292461384080417619968y^{8}z^{18}+587206670927131085830065487872y^{6}z^{20}-25372453915525139943201750122496y^{4}z^{22}-57302359991614086367621939200y^{2}z^{24}+288803894357734995292814573568z^{26})}$

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
12.48.0.c.4 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.ik.1 $24$ $2$ $2$ $1$ $1$ 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
24.192.5.ex.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ey.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ez.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fa.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fn.4 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.fo.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fp.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.fq.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.bi.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.m.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ei.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ej.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.wj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wl.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wm.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wz.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xa.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.xc.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.wj.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wk.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wl.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wm.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wz.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xa.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xb.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.xc.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.wj.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wk.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wl.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wm.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wz.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xa.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xb.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.xc.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.wj.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wk.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wl.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wm.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wz.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xa.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xb.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.xc.4 $312$ $2$ $2$ $5$ $?$ not computed