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Properties

Label 24.192.1-24.bl.2.1
Level $24$
Index $192$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $576$
Index: $192$ $\PSL_2$-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 $2^{2}\cdot4^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-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: 24.192.1.482

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&16\\20&7\end{bmatrix}$, $\begin{bmatrix}7&18\\4&1\end{bmatrix}$, $\begin{bmatrix}17&16\\8&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $D_4\times \GL(2,3)$
Contains $-I$: no $\quad$ (see 24.96.1.bl.2 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 36x $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

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{2^2}{3^4}\cdot\frac{708234904x^{2}y^{62}-267952361920x^{2}y^{61}z-98725924056896x^{2}y^{60}z^{2}+63666419509877760x^{2}y^{59}z^{3}-7871221345523189760x^{2}y^{58}z^{4}-4551733399496756060160x^{2}y^{57}z^{5}-256643220174983451537408x^{2}y^{56}z^{6}+90633885982123047298007040x^{2}y^{55}z^{7}+19131969588947354194393300992x^{2}y^{54}z^{8}+1628512750762356182959002746880x^{2}y^{53}z^{9}+53825148347243129803677110501376x^{2}y^{52}z^{10}-1407412960168201686009193105981440x^{2}y^{51}z^{11}-47805397689507224649508296497037312x^{2}y^{50}z^{12}+13672953169036479747409196772847779840x^{2}y^{49}z^{13}+1179860162370346439757128876192188858368x^{2}y^{48}z^{14}-1618506922212152467719544776950069506080768x^{2}y^{46}z^{16}-80007648852027960467638238173008449293516800x^{2}y^{45}z^{17}+4676089427213140968031368665537755848810430464x^{2}y^{44}z^{18}+42343481801714787591939431801148426130733137920x^{2}y^{43}z^{19}+4996777051103150051639113074420459107183807692800x^{2}y^{42}z^{20}-395795787553359743701990721564623831302818444083200x^{2}y^{41}z^{21}+4459090683439368154601067138603286720241920125173760x^{2}y^{40}z^{22}-111557538167462003138485574353480336744611919272345600x^{2}y^{39}z^{23}+10232484797138444661116981447996605265182763613916495872x^{2}y^{38}z^{24}-120130296093418322793350022742390605261261921596945203200x^{2}y^{37}z^{25}-890469179240101743681549752281652990267484936024782536704x^{2}y^{36}z^{26}-76762278669657020018193861067680126524261958851192904744960x^{2}y^{35}z^{27}+1093446812977761224728505054484727703644227212695978452713472x^{2}y^{34}z^{28}+39128853386360737397641177082024024916502241074092966307430400x^{2}y^{33}z^{29}-131944393763116613235479548102812948765781403332578228779876352x^{2}y^{32}z^{30}-9128708620822805288972326689694161121275084564228936713298247680x^{2}y^{31}z^{31}-231912952825487546783900296007749772919472567370891921870945255424x^{2}y^{30}z^{32}+4022230811825866946913530622716941941867999711875238136906902405120x^{2}y^{29}z^{33}+64136149210340089194192431053916300124253457968890877735620864638976x^{2}y^{28}z^{34}-359810907896764269256169956283894340762233731434685377853704791654400x^{2}y^{27}z^{35}-18377588790436903929361288581936280075369343025628116121258876257435648x^{2}y^{26}z^{36}-83434995468816353949249540973760732568599722277198975792064855067852800x^{2}y^{25}z^{37}+5437302154690757491753804501456376733916436733993118979936029725469704192x^{2}y^{24}z^{38}+10381239474304742369305895278656245147907699442384784602804200644857036800x^{2}y^{23}z^{39}-758805764521426129398325075047723871554393562883298953555388228202244603904x^{2}y^{22}z^{40}-4800000219415093747977763916772021481916011042519886598577426220578473246720x^{2}y^{21}z^{41}+123019903703423982955465752586578312355268608539361717770881537629237844901888x^{2}y^{20}z^{42}+932538835340975151814670225992876853855699130077244729537351586871098705182720x^{2}y^{19}z^{43}-23096058827662577137991077052547357772807466175824943649073337367354361379815424x^{2}y^{18}z^{44}-2401932720388465781600600456589806963536981678581503810335495338284227818946560x^{2}y^{17}z^{45}+1816181028744167507582971581592104867083474225770866515985943616860858705363599360x^{2}y^{16}z^{46}+635507795825853037260283764873362425372565560355010295598288127529189979644231680x^{2}y^{15}z^{47}-174199402264142579491005704113525988763993113992147265219980951515620469835611242496x^{2}y^{14}z^{48}+224597997868098189937649309586233851923412149798296992410434187342889465763604725760x^{2}y^{13}z^{49}+16990895427079662615072066117509565823608481283915327946208559198114123164542534942720x^{2}y^{12}z^{50}-144985965972741083790458006908101560583879281427626755664889391088480080196157875159040x^{2}y^{11}z^{51}+368648614674315329034866071022785977263678107484787681722572703942494726761946565050368x^{2}y^{10}z^{52}-1490239168276894181185164630141734244293299201731353813640754428799835157421291827363840x^{2}y^{9}z^{53}+43629626978755605812710383728541909614722375357115021777622431635407747243977125518114816x^{2}y^{8}z^{54}-446859149332690404171365325680066687211688745529474867347541490304098096307187431956807680x^{2}y^{7}z^{55}+2197234009276657084860577171861346429597450785910373473111186874935211436180136841839640576x^{2}y^{6}z^{56}-6042858687103105979456104704353416766935392409769118597487862831489200442993735877305303040x^{2}y^{5}z^{57}+17354164892561692958997352115090190480786587546808991843880517233576309364284776791316365312x^{2}y^{4}z^{58}-109242966240564231005154676973270384004404265611349644775118260121936974456497392214731653120x^{2}y^{3}z^{59}+538480103956807300456013565917476853265396014974752215731527300783142924948137854735065546752x^{2}y^{2}z^{60}-1388601098435051551805434885359784693402223370860504401685419178533049073001161453414918389760x^{2}yz^{61}+1533295666675326295794451246981664255121676710047659850564971590695241639439731640228600872960x^{2}z^{62}+25366040xy^{63}+66339843280xy^{62}z-59383795288960xy^{61}z^{2}+4333495235014656xy^{60}z^{3}+2565153994349905920xy^{59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Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.0-8.e.2.4 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-8.e.2.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.d.1.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.d.1.9 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.t.1.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.t.1.5 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.v.2.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.v.2.11 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.1-24.bb.1.4 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.bb.1.5 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.bg.1.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.bg.1.3 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.bi.1.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.bi.1.5 $24$ $2$ $2$ $1$ $1$ 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
24.576.17-24.st.1.1 $24$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
24.768.17-24.ho.1.5 $24$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$