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Properties

Label 24.96.1.cr.2
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: 24.96.1.1602

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&22\\12&11\end{bmatrix}$, $\begin{bmatrix}13&13\\0&11\end{bmatrix}$, $\begin{bmatrix}13&18\\0&7\end{bmatrix}$, $\begin{bmatrix}23&15\\0&19\end{bmatrix}$, $\begin{bmatrix}23&16\\0&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035859
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cr.2.1, 24.192.1-24.cr.2.2, 24.192.1-24.cr.2.3, 24.192.1-24.cr.2.4, 24.192.1-24.cr.2.5, 24.192.1-24.cr.2.6, 24.192.1-24.cr.2.7, 24.192.1-24.cr.2.8, 24.192.1-24.cr.2.9, 24.192.1-24.cr.2.10, 24.192.1-24.cr.2.11, 24.192.1-24.cr.2.12, 24.192.1-24.cr.2.13, 24.192.1-24.cr.2.14, 24.192.1-24.cr.2.15, 24.192.1-24.cr.2.16, 120.192.1-24.cr.2.1, 120.192.1-24.cr.2.2, 120.192.1-24.cr.2.3, 120.192.1-24.cr.2.4, 120.192.1-24.cr.2.5, 120.192.1-24.cr.2.6, 120.192.1-24.cr.2.7, 120.192.1-24.cr.2.8, 120.192.1-24.cr.2.9, 120.192.1-24.cr.2.10, 120.192.1-24.cr.2.11, 120.192.1-24.cr.2.12, 120.192.1-24.cr.2.13, 120.192.1-24.cr.2.14, 120.192.1-24.cr.2.15, 120.192.1-24.cr.2.16, 168.192.1-24.cr.2.1, 168.192.1-24.cr.2.2, 168.192.1-24.cr.2.3, 168.192.1-24.cr.2.4, 168.192.1-24.cr.2.5, 168.192.1-24.cr.2.6, 168.192.1-24.cr.2.7, 168.192.1-24.cr.2.8, 168.192.1-24.cr.2.9, 168.192.1-24.cr.2.10, 168.192.1-24.cr.2.11, 168.192.1-24.cr.2.12, 168.192.1-24.cr.2.13, 168.192.1-24.cr.2.14, 168.192.1-24.cr.2.15, 168.192.1-24.cr.2.16, 264.192.1-24.cr.2.1, 264.192.1-24.cr.2.2, 264.192.1-24.cr.2.3, 264.192.1-24.cr.2.4, 264.192.1-24.cr.2.5, 264.192.1-24.cr.2.6, 264.192.1-24.cr.2.7, 264.192.1-24.cr.2.8, 264.192.1-24.cr.2.9, 264.192.1-24.cr.2.10, 264.192.1-24.cr.2.11, 264.192.1-24.cr.2.12, 264.192.1-24.cr.2.13, 264.192.1-24.cr.2.14, 264.192.1-24.cr.2.15, 264.192.1-24.cr.2.16, 312.192.1-24.cr.2.1, 312.192.1-24.cr.2.2, 312.192.1-24.cr.2.3, 312.192.1-24.cr.2.4, 312.192.1-24.cr.2.5, 312.192.1-24.cr.2.6, 312.192.1-24.cr.2.7, 312.192.1-24.cr.2.8, 312.192.1-24.cr.2.9, 312.192.1-24.cr.2.10, 312.192.1-24.cr.2.11, 312.192.1-24.cr.2.12, 312.192.1-24.cr.2.13, 312.192.1-24.cr.2.14, 312.192.1-24.cr.2.15, 312.192.1-24.cr.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

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

Models

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

$ 0 $ $=$ $ 4 x^{2} + 2 x y + 2 x z - 2 y^{2} + 2 y z + z^{2} $
$=$ $10 x^{2} - 16 x y + 8 x z + 4 y^{2} - 4 y z - 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 180 x^{4} + 12 x^{3} y + 2 x^{2} y^{2} + 24 x^{2} z^{2} + 2 x y z^{2} - 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 6z$
$\displaystyle Z$ $=$ $\displaystyle w$

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{2^4}{5^6}\cdot\frac{50213679560958612827700405288xz^{23}-711527152195941448455178176024xz^{21}w^{2}+3164545295478415367527172463000xz^{19}w^{4}-3445835612394703249373214351600xz^{17}w^{6}-4044232935332923105227872727000xz^{15}w^{8}-1372805785844367682637311650000xz^{13}w^{10}-204862918122908892112577400000xz^{11}w^{12}-12536749175056250650852500000xz^{9}w^{14}-77094046607394176836875000xz^{7}w^{16}+11763900761991692934375000xz^{5}w^{18}-70996237199538046875000xz^{3}w^{20}-1378997874711562500000xzw^{22}+9774310281815253095610906672y^{2}z^{22}-139817927474289362349580479660y^{2}z^{20}w^{2}+634900306926466822986016654500y^{2}z^{18}w^{4}-757432101562462220805740544750y^{2}z^{16}w^{6}-679672616281845851807669925000y^{2}z^{14}w^{8}-182842675070720775721787925000y^{2}z^{12}w^{10}-20301007751617615404538500000y^{2}z^{10}w^{12}-723282280124313508790625000y^{2}z^{8}w^{14}+14343354551918212537500000y^{2}z^{6}w^{16}+487348229673843398437500y^{2}z^{4}w^{18}-11706915613251445312500y^{2}z^{2}w^{20}+109319937742089843750y^{2}w^{22}-47159805266060553388579656672yz^{23}+671487863075689286320182099432yz^{21}w^{2}-3018575441111231319958462427100yz^{19}w^{4}+3449676887537688198922572978300yz^{17}w^{6}+3532118627134067366479740201000yz^{15}w^{8}+1084498995837627083721557025000yz^{13}w^{10}+145624311176388407994271950000yz^{11}w^{12}+7749902333136808075061250000yz^{9}w^{14}+7129653691518792551250000yz^{7}w^{16}-6846004382423887209375000yz^{5}w^{18}+59323771376857617187500yz^{3}w^{20}+470859061871601562500yzw^{22}+46182380740628380379684781039z^{24}-651295350395761540307964840864z^{22}w^{2}+2866302089710121214442930317435z^{20}w^{4}-2971351985631617059723787831550z^{18}w^{6}-3941508989171940688010658903750z^{16}w^{8}-1503344799271941136534887525000z^{14}w^{10}-262731507889612990930003462500z^{12}w^{12}-21064469306296146819663375000z^{10}w^{14}-491870338696019685184453125z^{8}w^{16}+16205524655455087312500000z^{6}w^{18}+269929313512066142578125z^{4}w^{20}-8317717326540175781250z^{2}w^{22}+72896647795263671875w^{24}}{w^{2}(1148135074969482421875000xz^{21}+2613371857613525390625000xz^{19}w^{2}+2635331294276035743179664xz^{17}w^{4}+1543323637480722131905848xz^{15}w^{6}+577574454059343532752840xz^{13}w^{8}+143000944150378558575600xz^{11}w^{10}+23389508744185355280000xz^{9}w^{12}+2439197195567353312500xz^{7}w^{14}+149101861014787950000xz^{5}w^{16}+4454800879709250000xz^{3}w^{18}+41020293120000000xzw^{20}+223489688845825195312500y^{2}z^{20}+478616760763549804687500y^{2}z^{18}w^{2}+450462532560425231370966y^{2}z^{16}w^{4}+243665917504723685010600y^{2}z^{14}w^{6}+83059848575327931219900y^{2}z^{12}w^{8}+18368075167688498575500y^{2}z^{10}w^{10}+2606600646433160043750y^{2}z^{8}w^{12}+225109044905495962500y^{2}z^{6}w^{14}+10477857527690625000y^{2}z^{4}w^{16}+198724402248750000y^{2}z^{2}w^{18}+629145600000000y^{2}w^{20}-1078308947697143554687500yz^{21}-2380499129855346679687500yz^{19}w^{2}-2322058268059387633714716yz^{17}w^{4}-1311205095941729467096584yz^{15}w^{6}-471263388165738985960620yz^{13}w^{8}-111489012729225524796300yz^{11}w^{10}-17309030856099354765000yz^{9}w^{12}-1698195767308143412500yz^{7}w^{14}-96468966948047850000yz^{5}w^{16}-2634676342310250000yz^{3}w^{18}-21768437760000000yzw^{20}+1055959978812561035156250z^{22}+2474644126817779541015625z^{20}w^{2}+2582448379589080597205142z^{18}w^{4}+1575627666812594858090718z^{16}w^{6}+619931290458158869898070z^{14}w^{8}+163438243546817445792525z^{12}w^{10}+29008489288525668544500z^{10}w^{12}+3382896571164767784375z^{8}w^{14}+243760487014610606250z^{6}w^{16}+9560664402109312500z^{4}w^{18}+154050202314375000z^{2}w^{20}+419430400000000w^{22})}$

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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.cg.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.dh.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.di.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dj.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dk.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dl.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dm.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dn.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.do.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.i.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.i.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.be.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.bf.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.rf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.rg.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.rh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ri.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.rn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ro.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.rp.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.rq.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.rf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.rg.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.rh.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ri.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.rn.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ro.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.rp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.rq.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.rf.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.rg.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.rh.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ri.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.rn.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ro.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.rp.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.rq.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.rf.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.rg.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.rh.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ri.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.rn.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ro.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.rp.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.rq.2 $312$ $2$ $2$ $5$ $?$ not computed