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Properties

Label 24.288.5-24.fd.1.5
Level $24$
Index $288$
Genus $5$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $576$
Index: $288$ $\PSL_2$-index:$144$
Genus: $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $3^{8}\cdot6^{4}\cdot24^{4}$ Cusp orbits $1^{4}\cdot2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24R5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.288.5.1027

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&3\\0&5\end{bmatrix}$, $\begin{bmatrix}11&15\\0&13\end{bmatrix}$, $\begin{bmatrix}13&18\\12&19\end{bmatrix}$, $\begin{bmatrix}17&21\\0&13\end{bmatrix}$, $\begin{bmatrix}19&15\\12&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4:D_8$
Contains $-I$: no $\quad$ (see 24.144.5.fd.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $256$

Jacobian

Conductor: $2^{26}\cdot3^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 36.2.a.a, 192.2.a.d$^{2}$, 576.2.a.d, 576.2.a.e

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

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

$ 0 $ $=$ $ x^{6} y - 2 x^{6} z - 5 x^{4} y^{2} z + 8 x^{4} y z^{2} + 4 x^{4} z^{3} + 8 x^{2} y^{3} z^{2} + \cdots + 4 y^{2} z^{5} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1:0:0)$, $(0:-1:-1:2:1)$, $(0:1:0:0:0)$, $(0:1:1:2:1)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^{10}\cdot3^2}\cdot\frac{21134460321792y^{18}+15850845241344y^{15}t^{3}+11888133931008y^{14}t^{4}+2972033482752y^{13}t^{5}-8420761534464y^{12}t^{6}-10773621374976y^{11}t^{7}-3668603830272y^{10}t^{8}-1438288773120y^{9}t^{9}-7575202529280y^{8}t^{10}-13308847202304y^{7}t^{11}-6462745823232y^{6}t^{12}+10443203956224y^{5}t^{13}+20153966006016y^{4}t^{14}+13338328242240y^{3}t^{15}+1056216312684y^{2}t^{16}+548394145047yt^{17}+21134460321792z^{18}+126806761930752z^{15}t^{3}+760840571584512z^{14}t^{4}+4945463715299328z^{13}t^{5}+33413581768753152z^{12}t^{6}+235860577191198720z^{11}t^{7}+1719119271495204864z^{10}t^{8}+12839607334695075840z^{9}t^{9}+97764399455894765568z^{8}t^{10}+756132109707261247488z^{7}t^{11}+5924049757292659212288z^{6}t^{12}+46918151040067977609216z^{5}t^{13}+375025994234952966733824z^{4}t^{14}+3021502159822867490930688z^{3}t^{15}+24511878840794799843311616z^{2}t^{16}-2819417274777600zw^{17}+97200739219046400zw^{16}t-1665711452165013504zw^{15}t^{2}+18899120069850341376zw^{14}t^{3}-159656408040639799296zw^{13}t^{4}+1071123904061666703360zw^{12}t^{5}-5944683902059184689152zw^{11}t^{6}+28067472041129554346496zw^{10}t^{7}-115011141958073922324480zw^{9}t^{8}+414920222840872022046336zw^{8}t^{9}-1330785464886391042222848zw^{7}t^{10}+3814533696472887733693920zw^{6}t^{11}-9767849763835681844693472zw^{5}t^{12}+22132713885815371033301832zw^{4}t^{13}-43111950613760560503975696zw^{3}t^{14}+65817934701310552497279678zw^{2}t^{15}-52514941697497555401727827zwt^{16}+1001982106055235642433623zt^{17}+1289663125258240w^{18}-44895535853469696w^{17}t+778076478814519296w^{16}t^{2}-8941805528681201664w^{15}t^{3}+76629301237313593344w^{14}t^{4}-522300932927781949440w^{13}t^{5}+2949253273090661332992w^{12}t^{6}-14187560595182210589696w^{11}t^{7}+59319122252417217833472w^{10}t^{8}-218701419172094417129728w^{9}t^{9}+718191378087021553468032w^{8}t^{10}-2113072490113588234017216w^{7}t^{11}+5575654131030061933473312w^{6}t^{12}-13110405620788696796242992w^{5}t^{13}+26880354591657269407024392w^{4}t^{14}-45484893990975283565444796w^{3}t^{15}+45568521438516208483243344w^{2}t^{16}-30358113389151200755263288wt^{17}+8129174934659294287088164t^{18}}{559872y^{6}t^{12}+1679616y^{5}t^{13}+2309472y^{4}t^{14}+1819584y^{3}t^{15}+1003833y^{2}t^{16}+1082565yt^{17}+2293235712z^{6}t^{12}+55037657088z^{5}t^{13}+853083684864z^{4}t^{14}+10874523746304z^{3}t^{15}+124206232633344z^{2}t^{16}-20561920zw^{17}+426606592zw^{16}t-4308348928zw^{15}t^{2}+28329095168zw^{14}t^{3}-136905070592zw^{13}t^{4}+520665411584zw^{12}t^{5}-1630439711232zw^{11}t^{6}+4346412799488zw^{10}t^{7}-10128210036288zw^{9}t^{8}+21091464566144zw^{8}t^{9}-40000484327888zw^{7}t^{10}+70217803495568zw^{6}t^{11}-115651095535204zw^{5}t^{12}+180479903966320zw^{4}t^{13}-266210238079687zw^{3}t^{14}+348289770989277zw^{2}t^{15}-261726989590755zwt^{16}+3618726616197zt^{17}+9404416w^{18}-198295552w^{17}t+2045009920w^{16}t^{2}-13798658048w^{15}t^{3}+68750311424w^{14}t^{4}-270697281536w^{13}t^{5}+880604611584w^{12}t^{6}-2444444321280w^{11}t^{7}+5937738654336w^{10}t^{8}-12885441267776w^{9}t^{9}+25424657405600w^{8}t^{10}-46309121233328w^{7}t^{11}+78877660671352w^{6}t^{12}-126908358501844w^{5}t^{13}+193171301206006w^{4}t^{14}-269351825437845w^{3}t^{15}+248470306717068w^{2}t^{16}-157625812167642wt^{17}+41307375501747t^{18}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.5.fd.1 :

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

Equation of the image curve:

$0$ $=$ $ X^{6}Y-2X^{6}Z-5X^{4}Y^{2}Z+8X^{4}YZ^{2}+4X^{4}Z^{3}+8X^{2}Y^{3}Z^{2}-8X^{2}Y^{2}Z^{3}-4X^{2}YZ^{4}-4Y^{4}Z^{3}+4Y^{2}Z^{5} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(3)$ $3$ $24$ $12$ $0$ $0$ full Jacobian
8.24.0-8.p.1.7 $8$ $12$ $12$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.96.1-24.ix.1.22 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.144.1-12.f.1.4 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.144.1-12.f.1.16 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.144.3-24.pc.1.9 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.144.3-24.pc.1.21 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.144.3-24.pp.1.13 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.144.3-24.pp.1.15 $24$ $2$ $2$ $3$ $1$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.576.9-24.bb.1.2 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.576.9-24.bb.2.3 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.576.9-24.bd.1.3 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.576.9-24.bd.2.3 $24$ $2$ $2$ $9$ $1$ $2^{2}$
24.576.13-24.ea.1.2 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.et.1.12 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.jk.1.1 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.jm.1.10 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.lh.1.7 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.576.13-24.lh.2.7 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.576.13-24.lj.1.7 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.576.13-24.lj.2.7 $24$ $2$ $2$ $13$ $1$ $2^{4}$
24.576.13-24.nv.1.5 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.ny.1.5 $24$ $2$ $2$ $13$ $2$ $1^{8}$
24.576.13-24.oi.1.3 $24$ $2$ $2$ $13$ $3$ $1^{8}$
24.576.13-24.oo.1.5 $24$ $2$ $2$ $13$ $3$ $1^{8}$