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Properties

Label 24.192.1-24.cj.1.9
Level $24$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $192$
Index: $192$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.1.1661

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&16\\12&5\end{bmatrix}$, $\begin{bmatrix}13&6\\0&19\end{bmatrix}$, $\begin{bmatrix}17&20\\18&5\end{bmatrix}$, $\begin{bmatrix}23&14\\18&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $S_3\times C_2^3:D_4$
Contains $-I$: no $\quad$ (see 24.96.1.cj.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x z + 2 x w + y z - y w $
$=$ $3 x^{2} - 6 x y + 6 y^{2} + 2 z^{2} + 8 z w + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 8 x^{3} z + 30 x^{2} y^{2} + 21 x^{2} z^{2} + 48 x y^{2} z + 20 x z^{3} + 30 y^{2} z^{2} + 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^2\cdot3^2\cdot5}\cdot\frac{4751452803611755371093750000xy^{21}w^{2}+33222158002853393554687500000xy^{19}w^{4}-374747564101409912109375000000xy^{17}w^{6}-1489131201966394042968750000000xy^{15}w^{8}-1280617439279361328125000000000xy^{13}w^{10}+35349859100682330243750000000000xy^{11}w^{12}+75407888216459532147300000000000xy^{9}w^{14}-389487685642820917764177600000000xy^{7}w^{16}+245695288458562247807671795200000xy^{5}w^{18}+113671810773543359832067169280000xy^{3}w^{20}+296388434325785078864123298791424xyw^{22}-494943000376224517822265625y^{24}-2375726401805877685546875000y^{22}w^{2}-4922505104541778564453125000y^{20}w^{4}+625958977138137817382812500000y^{18}w^{6}+1122742802743519592285156250000y^{16}w^{8}-11089684164914499023437500000000y^{14}w^{10}-24440744791493992223437500000000y^{12}w^{12}-28371887240106142587150000000000y^{10}w^{14}+532562500164773633695863300000000y^{8}w^{16}-991413230790007647596816121600000y^{6}w^{18}+1036152631751890065166564104960000y^{4}w^{20}-1230194562982395870286449077870592y^{2}w^{22}-8003000375000000000000000000z^{24}-86378390548125000000000000000z^{23}w-459175373765625000000000000000z^{22}w^{2}-1873621906653500000000000000000z^{21}w^{3}-6911014884126645000000000000000z^{20}w^{4}-21867372071717550000000000000000z^{19}w^{5}-58186583848750614200000000000000z^{18}w^{6}-138777797715893145360000000000000z^{17}w^{7}-302040295033939680744000000000000z^{16}w^{8}-567242472161990262670400000000000z^{15}w^{9}-907728119933099951466240000000000z^{14}w^{10}-1305384281226222957329664000000000z^{13}w^{11}-1629889851776752238090470400000000z^{12}w^{12}-1455071033545313206925199360000000z^{11}w^{13}-781519107989974740993825792000000z^{10}w^{14}-489814586779290430197957017600000z^{9}w^{15}-531085260978977577850915430400000z^{8}w^{16}-6686168464351076342460641280000z^{7}w^{17}-38016490274463204780239163392000z^{6}w^{18}-1102568547637245702646801239244800z^{5}w^{19}-1930893419253677335686969208504320z^{4}w^{20}-5036810648119675577619939779084288z^{3}w^{21}-8796336967843222549103924035289088z^{2}w^{22}-5933061964701937695928196514250752zw^{23}-1046652979275584957449093767335936w^{24}}{422351360321044921875000xy^{19}w^{4}-2874241790771484375000000xy^{17}w^{6}-2495598728027343750000000xy^{15}w^{8}+54981880181718750000000000xy^{13}w^{10}-179784020160386718750000000xy^{11}w^{12}+302964133966044480000000000xy^{9}w^{14}-291901257457685498880000000xy^{7}w^{16}+128094339857436222382080000xy^{5}w^{18}+33564639947416755565363200xy^{3}w^{20}-14300036811963028174012416xyw^{22}-299165546894073486328125y^{20}w^{4}+5508588008880615234375000y^{18}w^{6}-18103821881561279296875000y^{16}w^{8}+2022544445053710937500000y^{14}w^{10}+127021250731108886718750000y^{12}w^{12}-423989117410164840000000000y^{10}w^{14}+853781525402858375040000000y^{8}w^{16}-1353482478388447564400640000y^{6}w^{18}+1901380874235553591219814400y^{4}w^{20}-2471855416004540665952796672y^{2}w^{22}-4613203125000000000000z^{24}-124556484375000000000000z^{23}w-1458002847656250000000000z^{22}w^{2}-9608287204687500000000000z^{21}w^{3}-38441424903875000000000000z^{20}w^{4}-91611573184587500000000000z^{19}w^{5}-104374745696930000000000000z^{18}w^{6}+51199146065734500000000000z^{17}w^{7}+354998075926146150000000000z^{16}w^{8}+491569869484909020000000000z^{15}w^{9}+176012028092382594000000000z^{14}w^{10}-438871162612751587200000000z^{13}w^{11}-779299751964355471920000000z^{12}w^{12}-409277681294847935744000000z^{11}w^{13}+231222135118487832131200000z^{10}w^{14}+645110389905853380463360000z^{9}w^{15}+864134570531749128766080000z^{8}w^{16}+554894753133696362347776000z^{7}w^{17}-72369139489891840020172800z^{6}w^{18}-2562074694762811928694528000z^{5}w^{19}-8257607484722539764378731520z^{4}w^{20}-12898439339392595289693124608z^{3}w^{21}-15069933825228548920504337408z^{2}w^{22}-11217304714156993974426693632zw^{23}-2132743666316271097756798976w^{24}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.cj.1 :

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ X^{4}+30X^{2}Y^{2}+8X^{3}Z+48XY^{2}Z+21X^{2}Z^{2}+30Y^{2}Z^{2}+20XZ^{3}+4Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.96.0-12.a.2.9 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-12.a.2.13 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.o.2.15 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.o.2.17 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.1-24.bw.1.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1-24.bw.1.5 $24$ $2$ $2$ $1$ $0$ 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
24.384.5-24.b.2.6 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.c.2.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.f.2.12 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.h.3.10 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.bm.4.5 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.bo.2.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.br.3.7 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.384.5-24.bt.2.5 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.576.9-24.b.1.6 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
120.384.5-120.ip.2.5 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ir.3.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.iu.3.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.iw.1.8 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.jg.3.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ji.4.10 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.jl.4.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.jn.3.6 $120$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.ip.1.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.ir.4.3 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.iu.1.9 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.iw.3.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.jg.2.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.ji.1.3 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.jl.1.4 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.jn.3.7 $168$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.ip.2.2 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.ir.3.3 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.iu.1.10 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.iw.3.12 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.jg.3.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.ji.1.3 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.jl.3.9 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.jn.2.9 $264$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.ip.3.12 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.ir.3.10 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.iu.4.5 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.iw.4.15 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.jg.2.8 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.ji.3.12 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.jl.1.15 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.jn.4.12 $312$ $2$ $2$ $5$ $?$ not computed