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Properties

Label 24.96.1.cj.1
Level $24$
Index $96$
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: $96$ $\PSL_2$-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: 24.96.1.1335

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&18\\0&17\end{bmatrix}$, $\begin{bmatrix}13&0\\6&17\end{bmatrix}$, $\begin{bmatrix}17&22\\0&13\end{bmatrix}$, $\begin{bmatrix}19&4\\12&11\end{bmatrix}$, $\begin{bmatrix}23&18\\18&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cj.1.1, 24.192.1-24.cj.1.2, 24.192.1-24.cj.1.3, 24.192.1-24.cj.1.4, 24.192.1-24.cj.1.5, 24.192.1-24.cj.1.6, 24.192.1-24.cj.1.7, 24.192.1-24.cj.1.8, 24.192.1-24.cj.1.9, 24.192.1-24.cj.1.10, 24.192.1-24.cj.1.11, 24.192.1-24.cj.1.12, 24.192.1-24.cj.1.13, 24.192.1-24.cj.1.14, 24.192.1-24.cj.1.15, 24.192.1-24.cj.1.16, 120.192.1-24.cj.1.1, 120.192.1-24.cj.1.2, 120.192.1-24.cj.1.3, 120.192.1-24.cj.1.4, 120.192.1-24.cj.1.5, 120.192.1-24.cj.1.6, 120.192.1-24.cj.1.7, 120.192.1-24.cj.1.8, 120.192.1-24.cj.1.9, 120.192.1-24.cj.1.10, 120.192.1-24.cj.1.11, 120.192.1-24.cj.1.12, 120.192.1-24.cj.1.13, 120.192.1-24.cj.1.14, 120.192.1-24.cj.1.15, 120.192.1-24.cj.1.16, 168.192.1-24.cj.1.1, 168.192.1-24.cj.1.2, 168.192.1-24.cj.1.3, 168.192.1-24.cj.1.4, 168.192.1-24.cj.1.5, 168.192.1-24.cj.1.6, 168.192.1-24.cj.1.7, 168.192.1-24.cj.1.8, 168.192.1-24.cj.1.9, 168.192.1-24.cj.1.10, 168.192.1-24.cj.1.11, 168.192.1-24.cj.1.12, 168.192.1-24.cj.1.13, 168.192.1-24.cj.1.14, 168.192.1-24.cj.1.15, 168.192.1-24.cj.1.16, 264.192.1-24.cj.1.1, 264.192.1-24.cj.1.2, 264.192.1-24.cj.1.3, 264.192.1-24.cj.1.4, 264.192.1-24.cj.1.5, 264.192.1-24.cj.1.6, 264.192.1-24.cj.1.7, 264.192.1-24.cj.1.8, 264.192.1-24.cj.1.9, 264.192.1-24.cj.1.10, 264.192.1-24.cj.1.11, 264.192.1-24.cj.1.12, 264.192.1-24.cj.1.13, 264.192.1-24.cj.1.14, 264.192.1-24.cj.1.15, 264.192.1-24.cj.1.16, 312.192.1-24.cj.1.1, 312.192.1-24.cj.1.2, 312.192.1-24.cj.1.3, 312.192.1-24.cj.1.4, 312.192.1-24.cj.1.5, 312.192.1-24.cj.1.6, 312.192.1-24.cj.1.7, 312.192.1-24.cj.1.8, 312.192.1-24.cj.1.9, 312.192.1-24.cj.1.10, 312.192.1-24.cj.1.11, 312.192.1-24.cj.1.12, 312.192.1-24.cj.1.13, 312.192.1-24.cj.1.14, 312.192.1-24.cj.1.15, 312.192.1-24.cj.1.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

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 between models of this curve

Birational map from embedded model to plane model:

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

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}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.a.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.bw.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.b.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.c.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.f.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.h.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bm.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bo.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.br.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bt.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.b.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.b.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.j.4 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.v.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.ip.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ir.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.iu.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.iw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ji.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jl.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jn.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.ip.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ir.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.iu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.iw.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jg.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ji.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jn.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.ip.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ir.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.iu.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.iw.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jg.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ji.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jl.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jn.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.ip.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ir.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.iu.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.iw.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jg.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ji.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jn.4 $312$ $2$ $2$ $5$ $?$ not computed