↑

Properties

Label 24.96.1.cn.4
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

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: $0$
$\Q$-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: 24.96.1.1334

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&14\\18&17\end{bmatrix}$, $\begin{bmatrix}11&4\\12&13\end{bmatrix}$, $\begin{bmatrix}11&12\\18&5\end{bmatrix}$, $\begin{bmatrix}11&14\\0&1\end{bmatrix}$, $\begin{bmatrix}19&4\\0&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cn.4.1, 24.192.1-24.cn.4.2, 24.192.1-24.cn.4.3, 24.192.1-24.cn.4.4, 24.192.1-24.cn.4.5, 24.192.1-24.cn.4.6, 24.192.1-24.cn.4.7, 24.192.1-24.cn.4.8, 24.192.1-24.cn.4.9, 24.192.1-24.cn.4.10, 24.192.1-24.cn.4.11, 24.192.1-24.cn.4.12, 24.192.1-24.cn.4.13, 24.192.1-24.cn.4.14, 24.192.1-24.cn.4.15, 24.192.1-24.cn.4.16, 120.192.1-24.cn.4.1, 120.192.1-24.cn.4.2, 120.192.1-24.cn.4.3, 120.192.1-24.cn.4.4, 120.192.1-24.cn.4.5, 120.192.1-24.cn.4.6, 120.192.1-24.cn.4.7, 120.192.1-24.cn.4.8, 120.192.1-24.cn.4.9, 120.192.1-24.cn.4.10, 120.192.1-24.cn.4.11, 120.192.1-24.cn.4.12, 120.192.1-24.cn.4.13, 120.192.1-24.cn.4.14, 120.192.1-24.cn.4.15, 120.192.1-24.cn.4.16, 168.192.1-24.cn.4.1, 168.192.1-24.cn.4.2, 168.192.1-24.cn.4.3, 168.192.1-24.cn.4.4, 168.192.1-24.cn.4.5, 168.192.1-24.cn.4.6, 168.192.1-24.cn.4.7, 168.192.1-24.cn.4.8, 168.192.1-24.cn.4.9, 168.192.1-24.cn.4.10, 168.192.1-24.cn.4.11, 168.192.1-24.cn.4.12, 168.192.1-24.cn.4.13, 168.192.1-24.cn.4.14, 168.192.1-24.cn.4.15, 168.192.1-24.cn.4.16, 264.192.1-24.cn.4.1, 264.192.1-24.cn.4.2, 264.192.1-24.cn.4.3, 264.192.1-24.cn.4.4, 264.192.1-24.cn.4.5, 264.192.1-24.cn.4.6, 264.192.1-24.cn.4.7, 264.192.1-24.cn.4.8, 264.192.1-24.cn.4.9, 264.192.1-24.cn.4.10, 264.192.1-24.cn.4.11, 264.192.1-24.cn.4.12, 264.192.1-24.cn.4.13, 264.192.1-24.cn.4.14, 264.192.1-24.cn.4.15, 264.192.1-24.cn.4.16, 312.192.1-24.cn.4.1, 312.192.1-24.cn.4.2, 312.192.1-24.cn.4.3, 312.192.1-24.cn.4.4, 312.192.1-24.cn.4.5, 312.192.1-24.cn.4.6, 312.192.1-24.cn.4.7, 312.192.1-24.cn.4.8, 312.192.1-24.cn.4.9, 312.192.1-24.cn.4.10, 312.192.1-24.cn.4.11, 312.192.1-24.cn.4.12, 312.192.1-24.cn.4.13, 312.192.1-24.cn.4.14, 312.192.1-24.cn.4.15, 312.192.1-24.cn.4.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 24x - 56 $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(2:0:1)$

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}{2^2\cdot3^2}\cdot\frac{48x^{2}y^{30}+228521088x^{2}y^{28}z^{2}+15973920783360x^{2}y^{26}z^{4}+45441092170358784x^{2}y^{24}z^{6}+28158764037753470976x^{2}y^{22}z^{8}+7589943998784641433600x^{2}y^{20}z^{10}+5227234686007865537200128x^{2}y^{18}z^{12}+5190720384101156302343897088x^{2}y^{16}z^{14}+2527906766975259541897375383552x^{2}y^{14}z^{16}+586380007106635225808974324432896x^{2}y^{12}z^{18}+35861867447704395158938474902454272x^{2}y^{10}z^{20}-13319988106684367996453743741372465152x^{2}y^{8}z^{22}-3452559991281036421772197206391441588224x^{2}y^{6}z^{24}-346982352921968316416905014983986614632448x^{2}y^{4}z^{26}-15660815939811320847590384391106651471478784x^{2}y^{2}z^{28}-241624017713473993193847791559410735778889728x^{2}z^{30}+26016xy^{30}z+6968353536xy^{28}z^{3}+330647042414592xy^{26}z^{5}+407106001258758144xy^{24}z^{7}+231354040818109906944xy^{22}z^{9}+51033650144785241997312xy^{20}z^{11}-15009306233279497843507200xy^{18}z^{13}-2855059496350392663039541248xy^{16}z^{15}+9035247526954379971342953873408xy^{14}z^{17}+6194396223517545788318262327508992xy^{12}z^{19}+1894669848429604148986696191770099712xy^{10}z^{21}+320643740653297143520376095657728933888xy^{8}z^{23}+30209909457827002325636276588873879912448xy^{6}z^{25}+1387929486759722595814030584918905462980608xy^{4}z^{27}+15660816147702595915688136614136384099188736xy^{2}z^{29}-483248035426947986387695583118821471557779456xz^{31}+y^{32}+1015680y^{30}z^{2}+717442748928y^{28}z^{4}+4644741343076352y^{26}z^{6}+3690683634913689600y^{24}z^{8}+1234640136688035692544y^{22}z^{10}+144779661158389175549952y^{20}z^{12}-283083354904458273478410240y^{18}z^{14}-320672256105110489055388237824y^{16}z^{16}-164526514670195150251637197504512y^{14}z^{18}-48092281871696101329380918509436928y^{12}z^{20}-8399386370648150236853895023346843648y^{10}z^{22}-847745885792173185503164469791520980992y^{8}z^{24}-41430704826740445052085252078804108574720y^{6}z^{26}-82861110641441567002286143438841356222464y^{4}z^{28}+71592303577447724575132922939079085183205376y^{2}z^{30}+1932992164160049652905339572562497010023792640z^{32}}{zy^{4}(y^{2}+216z^{2})^{2}(1008x^{2}y^{20}z-23763456x^{2}y^{18}z^{3}-105508438272x^{2}y^{16}z^{5}+9380480679936x^{2}y^{14}z^{7}+110881844172619776x^{2}y^{12}z^{9}+35944099864085790720x^{2}y^{10}z^{11}+1945756335583354945536x^{2}y^{8}z^{13}+101085468550861160448x^{2}y^{6}z^{15}-10917230603493005328384x^{2}y^{4}z^{17}-589530452588622287732736x^{2}y^{2}z^{19}-7958661109946400884391936x^{2}z^{21}-xy^{22}+46296xy^{20}z^{2}+310371264xy^{18}z^{4}-952170951168xy^{16}z^{6}-2678703355846656xy^{14}z^{8}-1195519612512043008xy^{12}z^{10}-89974606171423113216xy^{10}z^{12}+3895373574479416393728xy^{8}z^{14}-30009748476036907008xy^{6}z^{16}-37186816743148049399808xy^{4}z^{18}-1400134824897977933365248xy^{2}z^{20}-15917322219892801768783872xz^{22}-40y^{22}z-615024y^{20}z^{3}+6436599552y^{18}z^{5}+21709594432512y^{16}z^{7}+12568334875361280y^{14}z^{9}+435013266989776896y^{12}z^{11}-288565961040410443776y^{10}z^{13}-15457155661688003887104y^{8}z^{15}+1503646344694059761664y^{6}z^{17}-373232821256917119664128y^{4}z^{19}-17391148351364357488115712y^{2}z^{21}-222842511078499224762974208z^{23})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

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.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.by.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.bl.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bm.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bn.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bo.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bv.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bx.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.by.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ca.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.g.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.f.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.n.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.z.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.jx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jy.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ka.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ke.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kh.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ki.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.jx.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jy.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ka.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kb.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ke.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kh.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ki.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.jx.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jy.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ka.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kb.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ke.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kf.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kh.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ki.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.jx.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jy.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ka.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kb.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ke.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kf.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kh.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ki.4 $312$ $2$ $2$ $5$ $?$ not computed