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Properties

Label 24.96.1-24.cm.1.2
Level $24$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $576$
Index: $96$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{2}\cdot4$
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: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.745

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&11\\10&15\end{bmatrix}$, $\begin{bmatrix}21&23\\14&3\end{bmatrix}$, $\begin{bmatrix}23&5\\0&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $(C_2\times \GL(2,3)):D_4$
Contains $-I$: no $\quad$ (see 24.48.1.cm.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $768$

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 2 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^6}{3^4}\cdot\frac{503360x^{2}y^{30}-225544448x^{2}y^{29}z-341393291200x^{2}y^{28}z^{2}-122198935339008x^{2}y^{27}z^{3}-20754186428841984x^{2}y^{26}z^{4}-1695910169544081408x^{2}y^{25}z^{5}-64539250935353579520x^{2}y^{24}z^{6}+141366578984623296479232x^{2}y^{22}z^{8}+7679417478997618880151552x^{2}y^{21}z^{9}+153621109114108298874519552x^{2}y^{20}z^{10}-1095434507629672702993760256x^{2}y^{19}z^{11}-121011022824407557031099105280x^{2}y^{18}z^{12}-2321977977989661702323227852800x^{2}y^{17}z^{13}+9550749085726079800874008313856x^{2}y^{16}z^{14}+709439170167985600725782945071104x^{2}y^{15}z^{15}+8778191292738534319524742944522240x^{2}y^{14}z^{16}-100616395747203735427838894168604672x^{2}y^{13}z^{17}-1570485851014479275646755935975636992x^{2}y^{12}z^{18}-5428816927655438358030116697183092736x^{2}y^{11}z^{19}+264287014438140880529857005161407315968x^{2}y^{10}z^{20}+205511053608135934749754532539656044544x^{2}y^{9}z^{21}-7531255381420450157012597243092905492480x^{2}y^{8}z^{22}-136183842543904102039772418368723377717248x^{2}y^{7}z^{23}+1219322169697112674649628799118859218976768x^{2}y^{6}z^{24}-511860548314322383300482977232481909997568x^{2}y^{5}z^{25}-3103833674830425397408969050632103552614400x^{2}y^{4}z^{26}-176487498215794937955630142630523129252806656x^{2}y^{3}z^{27}+1015993707364765618469447455681350762108026880x^{2}y^{2}z^{28}-1976441103865416472801594185373521533101670400x^{2}yz^{29}+1307938731915605861273436665882851808378880000x^{2}z^{30}+13600xy^{31}-11247712xy^{30}z+13376733184xy^{29}z^{2}+11522665746432xy^{28}z^{3}+2972396250611712xy^{27}z^{4}+357377139724036608xy^{26}z^{5}+27611934193681956864xy^{25}z^{6}+1423265021810084413440xy^{24}z^{7}+33903921353431862476800xy^{23}z^{8}-624254557719704714477568xy^{22}z^{9}-66335358603478639967207424xy^{21}z^{10}-1861732342793954703471280128xy^{20}z^{11}-8821302269136717262645886976xy^{19}z^{12}+836339234608003277910728245248xy^{18}z^{13}+17377640594220014915361347469312xy^{17}z^{14}+30724018504024047225277765386240xy^{16}z^{15}-5764230865087918120954674743869440xy^{15}z^{16}-46274235936975253666640606760271872xy^{14}z^{17}+378423494000957475563549031521058816xy^{13}z^{18}+15740532134538452685373577222795821056xy^{12}z^{19}-30646512308141411196325632886777577472xy^{11}z^{20}-1113182798906106279351600262299865055232xy^{10}z^{21}-10508684935920827924518560405387818827776xy^{9}z^{22}+147561973504071202685513313604233276162048xy^{8}z^{23}+164204278707070978142856150307103022514176xy^{7}z^{24}-997892789096667017825488246105321053880320xy^{6}z^{25}-46573065261516767527989875648740384441368576xy^{5}z^{26}+282220548283273011371048299728018496825589760xy^{4}z^{27}-576461913363071886394704675174105393974476800xy^{3}z^{28}+399647971849511174456880231342921297690624000xy^{2}z^{29}+125y^{32}+6295552y^{31}z+5651772416y^{30}z^{2}+1718311108608y^{29}z^{3}+264612408003072y^{28}z^{4}+22932535907450880y^{27}z^{5}-394340260174331904y^{26}z^{6}-232521509578845782016y^{25}z^{7}-17613058386602967515136y^{24}z^{8}-621111283493017485312000y^{23}z^{9}-8031075449992445914251264y^{22}z^{10}+277124543728567167926403072y^{21}z^{11}+13650533494406148704240664576y^{20}z^{12}+164405945009211202056405123072y^{19}z^{13}-1962225292900442914893569458176y^{18}z^{14}-101597579857357782931915067621376y^{17}z^{15}-515979642438528061884197177917440y^{16}z^{16}+11609457570937410917909102661206016y^{15}z^{17}+289030510849390422120924178368430080y^{14}z^{18}-912028154720397258219646859543052288y^{13}z^{19}-25588159845502426774096748580201037824y^{12}z^{20}-208740289776762099966860208722524766208y^{11}z^{21}+3292586190854074168316593593640141455360y^{10}z^{22}+4678748702469826346292768071424638189568y^{9}z^{23}-25132307523500739587307100844170615455744y^{8}z^{24}-1157514708595911472897773874600602903773184y^{7}z^{25}+7055569590759486665173234776511988719681536y^{6}z^{26}-14487885686780259436692647480443057175789568y^{5}z^{27}+10091969250684186817332047832639338748837888y^{4}z^{28}+3029607251244242917137035368758899638272y^{3}z^{29}+8720346018189807046582020517957006786560y^{2}z^{30}+8481964022778388290699613091210566041600yz^{31}+2806532213419319655010901390474084352000z^{32}}{224x^{2}y^{30}+2747072x^{2}y^{28}z^{2}-210215452672x^{2}y^{26}z^{4}-4232306227415040x^{2}y^{24}z^{6}-9313357553499635712x^{2}y^{22}z^{8}-8226683749640073314304x^{2}y^{20}z^{10}-1391236848461211966111744x^{2}y^{18}z^{12}-575633572024553856401670144x^{2}y^{16}z^{14}+425625883977871372436228800512x^{2}y^{14}z^{16}-112301051552352087362939551481856x^{2}y^{12}z^{18}+27268585995016030566004553712402432x^{2}y^{10}z^{20}-4407850046251020499784875596337643520x^{2}y^{8}z^{22}+374743558005660555577092699039203328000x^{2}y^{6}z^{24}-16396575141993967132488987986427396489216x^{2}y^{4}z^{26}+452126888210639499636753476781274973601792x^{2}y^{2}z^{28}-10463509855324846890187493327062814467031040x^{2}z^{30}-15904xy^{30}z+320337920xy^{28}z^{3}-9756254499328xy^{26}z^{5}-63395972911595520xy^{24}z^{7}-114862881853693624320xy^{22}z^{9}-59623576822033403609088xy^{20}z^{11}-22409839967653314847309824xy^{18}z^{13}+6814815720903806463881773056xy^{16}z^{15}-2219866428662384765909373812736xy^{14}z^{17}+1042487213846523097671020285263872xy^{12}z^{19}-230645452024636981249338672049815552xy^{10}z^{21}+28362673065199728729359346590369710080xy^{8}z^{23}-2323679743886492713637336320598102507520xy^{6}z^{25}+125590835218239738054503375594159534505984xy^{4}z^{27}-3197183774796089395655041850743370381524992xy^{2}z^{29}-y^{32}+280064y^{30}z^{2}-4117090816y^{28}z^{4}-204743654129664y^{26}z^{6}-816654016818561024y^{24}z^{8}-758899498981244534784y^{22}z^{10}-358955024666456464293888y^{20}z^{12}+44387161076784297356034048y^{18}z^{14}-22940438850372173533317955584y^{16}z^{16}+16093673319902228037909570650112y^{14}z^{18}-3943282783443266460969464305287168y^{12}z^{20}+558518222583130666504187925133000704y^{10}z^{22}-53452327035648777202175374897289625600y^{8}z^{24}+3139772696190831897958877081090025062400y^{6}z^{26}-80737169431936516206473394688536776540160y^{4}z^{28}+3880637134604491374829888342383919104y^{2}z^{30}-22452257707354557240087211123792674816z^{32}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0-8.n.1.3 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0-8.n.1.1 $24$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.9-24.op.1.4 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.384.9-24.gt.1.6 $24$ $4$ $4$ $9$ $4$ $1^{8}$
120.480.17-120.iz.1.2 $120$ $5$ $5$ $17$ $?$ not computed