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Properties

Label 24.96.1.cu.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: $8$ Newform level: $288$
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 $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.716

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}15&23\\16&9\end{bmatrix}$, $\begin{bmatrix}17&11\\8&15\end{bmatrix}$, $\begin{bmatrix}23&9\\16&13\end{bmatrix}$, $\begin{bmatrix}23&18\\0&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1089384
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cu.2.1, 24.192.1-24.cu.2.2, 24.192.1-24.cu.2.3, 24.192.1-24.cu.2.4, 24.192.1-24.cu.2.5, 24.192.1-24.cu.2.6, 24.192.1-24.cu.2.7, 24.192.1-24.cu.2.8, 48.192.1-24.cu.2.1, 48.192.1-24.cu.2.2, 48.192.1-24.cu.2.3, 48.192.1-24.cu.2.4, 48.192.1-24.cu.2.5, 48.192.1-24.cu.2.6, 48.192.1-24.cu.2.7, 48.192.1-24.cu.2.8, 120.192.1-24.cu.2.1, 120.192.1-24.cu.2.2, 120.192.1-24.cu.2.3, 120.192.1-24.cu.2.4, 120.192.1-24.cu.2.5, 120.192.1-24.cu.2.6, 120.192.1-24.cu.2.7, 120.192.1-24.cu.2.8, 168.192.1-24.cu.2.1, 168.192.1-24.cu.2.2, 168.192.1-24.cu.2.3, 168.192.1-24.cu.2.4, 168.192.1-24.cu.2.5, 168.192.1-24.cu.2.6, 168.192.1-24.cu.2.7, 168.192.1-24.cu.2.8, 240.192.1-24.cu.2.1, 240.192.1-24.cu.2.2, 240.192.1-24.cu.2.3, 240.192.1-24.cu.2.4, 240.192.1-24.cu.2.5, 240.192.1-24.cu.2.6, 240.192.1-24.cu.2.7, 240.192.1-24.cu.2.8, 264.192.1-24.cu.2.1, 264.192.1-24.cu.2.2, 264.192.1-24.cu.2.3, 264.192.1-24.cu.2.4, 264.192.1-24.cu.2.5, 264.192.1-24.cu.2.6, 264.192.1-24.cu.2.7, 264.192.1-24.cu.2.8, 312.192.1-24.cu.2.1, 312.192.1-24.cu.2.2, 312.192.1-24.cu.2.3, 312.192.1-24.cu.2.4, 312.192.1-24.cu.2.5, 312.192.1-24.cu.2.6, 312.192.1-24.cu.2.7, 312.192.1-24.cu.2.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

$ 0 $ $=$ $ x^{2} + y^{2} - 2 y z - z^{2} $
$=$ $x^{2} + 6 x y - 18 x z + 2 x w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 37 x^{4} - 4 x^{3} y + 4 x^{3} z + 2 x^{2} y^{2} - 8 x^{2} y z + 6 x^{2} z^{2} - 4 x y z^{2} + 4 x z^{3} + 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}{37^8}\cdot\frac{1475461954013359797940974589658598349329792xz^{23}-695539330100622051723666046859967953061120xz^{22}w-798604036971106108382428776711544709320704xz^{21}w^{2}-152632635561530683974012622626146814870528xz^{20}w^{3}-86658795429702547508695558364308956225024xz^{19}w^{4}+12605551843501404607751480173206005508096xz^{18}w^{5}+35428023928469739975250298296277149347840xz^{17}w^{6}+13439484345328544587870059820396813897728xz^{16}w^{7}+4195270643191756470019572514903543795712xz^{15}w^{8}+314278724274168873875729022179758743552xz^{14}w^{9}+270239786745597555134621858960433414144xz^{13}w^{10}-675558634169052350009608908203281317888xz^{12}w^{11}-120400854451131781841726480188175794176xz^{11}w^{12}+189694320153130563977448752011930828800xz^{10}w^{13}-150661262203868533290598391776925777920xz^{9}w^{14}+26382686586472300373884999927070982144xz^{8}w^{15}+18712105912599208502463165137242521600xz^{7}w^{16}-11181873580728975801753109855097389056xz^{6}w^{17}+1761878934115844617299037508613439488xz^{5}w^{18}+502658579071812466065688855044685824xz^{4}w^{19}-176913683837557670566744635828207616xz^{3}w^{20}+6504101842875294396044666793885696xz^{2}w^{21}+2827068330414428121877640940355584xzw^{22}-47870482592181044432700057845760xw^{23}+15532421908822545610262119467380505395119992y^{2}z^{22}+1516446911628413414421357549755787343366944y^{2}z^{21}w+222079694935480605271655206517744549440704y^{2}z^{20}w^{2}+36136829271237030192195677727740553937536y^{2}z^{19}w^{3}-636625531398773452765692536324492052404736y^{2}z^{18}w^{4}-185987465519375891647181205890569583294592y^{2}z^{17}w^{5}-42238374488674538216302652525055116557632y^{2}z^{16}w^{6}-8887110575280355733191839181101494926336y^{2}z^{15}w^{7}+7998722462782843120453384987385502834432y^{2}z^{14}w^{8}+3867050263280668340709315672040720690176y^{2}z^{13}w^{9}+1418502164207553444516597368488414715904y^{2}z^{12}w^{10}+565459225144781758456612402609770786816y^{2}z^{11}w^{11}-175980563685631213571811072818217689088y^{2}z^{10}w^{12}+54123332193729763817541624979332362240y^{2}z^{9}w^{13}+13278461972252974884747721946440636416y^{2}z^{8}w^{14}-34025804878590633227198089935325741056y^{2}z^{7}w^{15}+7842471336732983648149302490632517632y^{2}z^{6}w^{16}+709773644484093234144183795185983488y^{2}z^{5}w^{17}-1111718013925979776947945651497975808y^{2}z^{4}w^{18}+152422577001521702417292178300633088y^{2}z^{3}w^{19}+22525881903237926485750060531187712y^{2}z^{2}w^{20}-4570590892285505493041062691241984y^{2}zw^{21}-127600599455897814939166706319360y^{2}w^{22}-82662578948960942953222661001898116201942024yz^{23}-3032893823256826828842715099511574686733888yz^{22}w-567114552705407860371724962173705627992224yz^{21}w^{2}-96282156257053322626362217642628131193856yz^{20}w^{3}+3416849920387847994783703400115332086890816yz^{19}w^{4}+788110482159733685351772333591514962772224yz^{18}w^{5}+160197875422165568083038550641159557223936yz^{17}w^{6}+31486107434184319584804700098106852497152yz^{16}w^{7}-45339614286538133247323999535911891769600yz^{15}w^{8}-19532179756351411404025802850404171655168yz^{14}w^{9}-6060513026052741939172259708867469696000yz^{13}w^{10}-1863985095947005231959541165792568303616yz^{12}w^{11}+349524893162975789379913872877845620736yz^{11}w^{12}-23637373682832042079213565829451472896yz^{10}w^{13}+12694074371347413661608871756070191104yz^{9}w^{14}+68573374422321869241827959995748835328yz^{8}w^{15}-10276260073103745898923097450729211904yz^{7}w^{16}-606737970528256934716771138680668160yz^{6}w^{17}+1559098497720300438675956563472080896yz^{5}w^{18}-158426754760232221662052139782242304yz^{4}w^{19}-30402713996405319024514637124648960yz^{3}w^{20}-1863458829908801843977214230462464yz^{2}w^{21}+280810947947361712542027149869056yzw^{22}+72589645340077328295476591001600yw^{23}+109035530032934894117067998091503948151042555z^{24}-1516446911628413414421357549755787343366944z^{23}w-345034857769927255100069755655961078551520z^{22}w^{2}-19160272707667409158028356868815400886048z^{21}w^{3}-4587953621970803179926600794646588304046304z^{20}w^{4}-828114911300769400446006309963290020735488z^{19}w^{5}-124919547362490556510947096806387805382464z^{18}w^{6}-19487120844704559626068233860333191223936z^{17}w^{7}+65757722444432749552478013561299948610384z^{16}w^{8}+25468870681139067844123509018597835644928z^{15}w^{9}+6340016427804137303383589693888939974656z^{14}w^{10}+1231146706246113613547325917482688738304z^{13}w^{11}+15258819547581545958706337087371769856z^{12}w^{12}-235348621202904734447849986652148752384z^{11}w^{13}-90168548291134265850811642063109302272z^{10}w^{14}-4425329708526131840894240668609679360z^{9}w^{15}-5995024000484464750351357835920279296z^{8}w^{16}-1825333498166489563472410841670279168z^{7}w^{17}-46997703007169185935183310522294272z^{6}w^{18}+358339422403719966001059751904174080z^{5}w^{19}-31968228632996562490794395103043584z^{4}w^{20}-18980632089901375276803703859314688z^{3}w^{21}+5652402182466453874646359510401024z^{2}w^{22}+433573953116170128949886491656192zw^{23}-29715698222515742875669606699008w^{24}}{z^{8}(1689079499644582760400xz^{15}-14978777959863064455360xz^{14}w+46634525277027562170080xz^{13}w^{2}-35192541613471245967296xz^{12}w^{3}+343336087051639736484802976xz^{11}w^{4}-94381239733360434144233856xz^{10}w^{5}-206642137197195323556239616xz^{9}w^{6}-75099546569991743232943872xz^{8}w^{7}-20246711909000993691262208xz^{7}w^{8}-4773569704506206260753920xz^{6}w^{9}-340306667516892749888512xz^{5}w^{10}+35655778653881332801536xz^{4}w^{11}-9188521332893884452864xz^{3}w^{12}+18243228933544041553920xz^{2}w^{13}+2911656703620748918784xzw^{14}-152904881911715463168xw^{15}+850015699127259028350y^{2}z^{14}-1246449120932454309220y^{2}z^{13}w-20097363473346077166738y^{2}z^{12}w^{2}+98802749326810419389480y^{2}z^{11}w^{3}+3615740489259633962719745712y^{2}z^{10}w^{4}+1059214499022371954835992088y^{2}z^{9}w^{5}+241386641805381303041601948y^{2}z^{8}w^{6}+50151942847172577657388864y^{2}z^{7}w^{7}+1829523523202138885751792y^{2}z^{6}w^{8}-1774247974960930824885952y^{2}z^{5}w^{9}-748754831026338763837248y^{2}z^{4}w^{10}-132465478697897392381440y^{2}z^{3}w^{11}-36258584820504005127936y^{2}z^{2}w^{12}-8863251590322091746304y^{2}zw^{13}-321637907045913987072y^{2}w^{14}-1700031398254518056700yz^{15}+2492898241864908618440yz^{14}w+40053970321721772436776yz^{13}w^{2}-196451104906945838005480yz^{12}w^{3}-19243381204918759454253829260yz^{11}w^{4}-4463896920335356841464112304yz^{10}w^{5}-912118297450703794701385440yz^{9}w^{6}-178580491543865672303260112yz^{8}w^{7}+13506470612777543555706672yz^{7}w^{8}+12041211161029641227741312yz^{6}w^{9}+3730206839678911069291584yz^{5}w^{10}+765007784398115201976576yz^{4}w^{11}+148994236052086822461696yz^{3}w^{12}+26009562361673323323392yz^{2}w^{13}+2582194664671332400128yzw^{14}+203483979219039805440yw^{15}-850015699127259028350z^{16}+1246449120932454309220z^{15}w+19956606848375695270038z^{14}w^{2}-97601436705145291317100z^{13}w^{3}+25383540978205726121803723863z^{12}w^{4}+4603249979215567131855343168z^{11}w^{5}+697458611402448924486804252z^{10}w^{6}+110098082903708831771081720z^{9}w^{7}-59815640287706760242551527z^{8}w^{8}-24054018503201168346840960z^{7}w^{9}-5831865266152015285289280z^{6}w^{10}-1233342543709906351130432z^{5}w^{11}-180237682495418656084704z^{4}w^{12}-16070909914990818690560z^{3}w^{13}+550821989210904761856z^{2}w^{14}-251151556876972166144zw^{15}-86671150897474228992w^{16})}$

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
8.48.0.n.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.be.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.dj.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
24.288.17.ckt.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.ru.1 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
48.192.5.gp.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.gr.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.gy.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hb.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hl.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ho.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hv.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hx.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
240.192.5.btb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.btd.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bup.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bur.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bwi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bwk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bxi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bxk.1 $240$ $2$ $2$ $5$ $?$ not computed