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Properties

Label 24.48.1.lw.1
Level $24$
Index $48$
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: $48$ $\PSL_2$-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 $4^{2}$
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: yes $\quad(D =$ $-12$)

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.297

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&10\\0&19\end{bmatrix}$, $\begin{bmatrix}13&2\\2&3\end{bmatrix}$, $\begin{bmatrix}23&21\\4&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $1536$

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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^4\cdot3^3\,\frac{157560x^{2}y^{30}-155218368x^{2}y^{29}z+19561399680x^{2}y^{28}z^{2}-694885727232x^{2}y^{27}z^{3}-6397939519488x^{2}y^{26}z^{4}+973806967222272x^{2}y^{25}z^{5}-24057115087288320x^{2}y^{24}z^{6}+14602037563825127424x^{2}y^{22}z^{8}-3133154050409082912768x^{2}y^{21}z^{9}+139484740764978483757056x^{2}y^{20}z^{10}+15837045622461598349131776x^{2}y^{19}z^{11}+894050610555672246217605120x^{2}y^{18}z^{12}+41020892225751061452187238400x^{2}y^{17}z^{13}+1357749378162926730891617107968x^{2}y^{16}z^{14}+34495712026190484049092553998336x^{2}y^{15}z^{15}+770850382390277403787505046650880x^{2}y^{14}z^{16}+15456645955156359346792372107214848x^{2}y^{13}z^{17}+257686167060920897506445278171889664x^{2}y^{12}z^{18}+3507714265943717949353434109649616896x^{2}y^{11}z^{19}+42667836873322039165585150989551271936x^{2}y^{10}z^{20}+508423597135622989197084241292483887104x^{2}y^{9}z^{21}+5558503175904129961282950131478807183360x^{2}y^{8}z^{22}+48414454272443811507176374422099845971968x^{2}y^{7}z^{23}+314703347791300269405623174939887634743296x^{2}y^{6}z^{24}+1679689660150934776467303431622393921011712x^{2}y^{5}z^{25}+9792021275932530977727291093973071613132800x^{2}y^{4}z^{26}+63298706701390221500321336685468559507193856x^{2}y^{3}z^{27}+318022823689590890905209275900257521610260480x^{2}y^{2}z^{28}+959155241581746229447832472313620744005222400x^{2}yz^{29}+1307938731915605861273436665882851808378880000x^{2}z^{30}-6600xy^{31}+21873264xy^{30}z-4386417024xy^{29}z^{2}+250721938944xy^{28}z^{3}-3169268011008xy^{27}z^{4}-94275641857536xy^{26}z^{5}-1074480292675584xy^{25}z^{6}+372671423330549760xy^{24}z^{7}-13339339859504332800xy^{23}z^{8}-609945082608618307584xy^{22}z^{9}+28106887613947648671744xy^{21}z^{10}+2778231296902714901397504xy^{20}z^{11}+255076503975682551456989184xy^{19}z^{12}+13253326707694572101560172544xy^{18}z^{13}+458012730160001841071105507328xy^{17}z^{14}+13355023679405485566141136896000xy^{16}z^{15}+340985453438375910138197688975360xy^{15}z^{16}+7081355888730267054826894292680704xy^{14}z^{17}+120928314495004905189757309025255424xy^{13}z^{18}+1878645083249336920370696403290161152xy^{12}z^{19}+27747749311594736406873282110237442048xy^{11}z^{20}+357163890220983706449632993308147974144xy^{10}z^{21}+3679650454857658619174299062648528961536xy^{9}z^{22}+31308240571764703149778612881716208992256xy^{8}z^{23}+258254613197898083168684021505195000201216xy^{7}z^{24}+2226074364832234122057217309126138307543040xy^{6}z^{25}+16703816576111563890403859058081745418059776xy^{5}z^{26}+88339696415115415749408713654534715431976960xy^{4}z^{27}+279753575602667238985665504128609970605260800xy^{3}z^{28}+399647971849511174456880231342921297690624000xy^{2}z^{29}+125y^{32}-2257952y^{31}z+877508992y^{30}z^{2}-73446838272y^{29}z^{3}+1605024585216y^{28}z^{4}+30899012198400y^{27}z^{5}-1994037896613888y^{26}z^{6}+76769671679852544y^{25}z^{7}-3504127259544403968y^{24}z^{8}+20722844081388257280y^{23}z^{9}-2962766319021685997568y^{22}z^{10}+555129242440831666225152y^{21}z^{11}+67036072412587465785212928y^{20}z^{12}+3213527828079001825116684288y^{19}z^{13}+118542630916695670625170096128y^{18}z^{14}+3667350024699392916265824681984y^{17}z^{15}+88600723330871976060261960253440y^{16}z^{16}+1724357735726419413456057651953664y^{15}z^{17}+30085389283074118498743839757434880y^{14}z^{18}+485000555833962291943416119313825792y^{13}z^{19}+6645749098815078487742863707281031168y^{12}z^{20}+72707158307423212174215116044577538048y^{11}z^{21}+668922124970398800973011829210585497600y^{10}z^{22}+6012513571691110525884418901493141209088y^{9}z^{23}+54474579629852844564524255404760286363648y^{8}z^{24}+415152379507417781599282051562941546758144y^{7}z^{25}+2208500706663019208389804766766437700206592y^{6}z^{26}+7030787422463408355949271776405162674880512y^{5}z^{27}+10092411995149987991715859105428796619096064y^{4}z^{28}-1086593797042966332934014886218062364672y^{3}z^{29}+2729612441644123486688380389401831669760y^{2}z^{30}-4116247246348335494015988706028657049600yz^{31}+2806532213419319655010901390474084352000z^{32}}{336x^{2}y^{30}-184171968x^{2}y^{28}z^{2}+1220082020352x^{2}y^{26}z^{4}-6020934591744000x^{2}y^{24}z^{6}+16187323230387044352x^{2}y^{22}z^{8}-25957855004174956560384x^{2}y^{20}z^{10}+25813646232228662832267264x^{2}y^{18}z^{12}-16512504632037190915225288704x^{2}y^{16}z^{14}+6016010467787345794414871052288x^{2}y^{14}z^{16}-1291587677795966077223466965139456x^{2}y^{12}z^{18}+172690571068173110213213059566010368x^{2}y^{10}z^{20}-14824050863053845338687942075476869120x^{2}y^{8}z^{22}+820243290358451419116437678003218022400x^{2}y^{6}z^{24}-29042862159646491111988436598593175945216x^{2}y^{4}z^{26}+678190332315959249455130215171912460402688x^{2}y^{2}z^{28}-10463509855324846890187493327062814467031040x^{2}z^{30}-48384xy^{30}z+5321272320xy^{28}z^{3}-29603866894848xy^{26}z^{5}+82893903018393600xy^{24}z^{7}-202787959603154780160xy^{22}z^{9}+308734452353132714262528xy^{20}z^{11}-275403898033900956385542144xy^{18}z^{13}+151162549254320369420262703104xy^{16}z^{15}-48595460003879117416201407430656xy^{14}z^{17}+9543289624394931352909025334263808xy^{12}z^{19}-1201927255216723199770714475700682752xy^{10}z^{21}+99906023179840469457220417288163819520xy^{8}z^{23}-5485254848424798258238143870288600760320xy^{6}z^{25}+188386252827359607081755063391239301758976xy^{4}z^{27}-3197183774796089395655041850743370381524992xy^{2}z^{29}-y^{32}+3856896y^{30}z^{2}-87719625216y^{28}z^{4}+581223296507904y^{26}z^{6}-1638540956966436864y^{24}z^{8}+2419107944562356649984y^{22}z^{10}-2525024368029662713479168y^{20}z^{12}+1780928642471926124922273792y^{18}z^{14}-724613954656143405487252045824y^{16}z^{16}+169721383507233821230986688462848y^{14}z^{18}-24350255802571022410639018901372928y^{12}z^{20}+2232014405950399327799386465156202496y^{10}z^{22}-131517286852199543207814846309256396800y^{8}z^{24}+4709694492901395299210880705389553254400y^{6}z^{26}-80737603608159137850955918094254095728640y^{4}z^{28}+5820955701906737062244832513575878656y^{2}z^{30}-22452257707354557240087211123792674816z^{32}}$

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
8.24.0.bq.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.cw.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dc.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.fb.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.dk.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.24.1.dt.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.24.1.eo.1 $24$ $2$ $2$ $1$ $1$ 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
24.144.9.ehu.1 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.192.9.rn.1 $24$ $4$ $4$ $9$ $4$ $1^{8}$
48.96.3.uz.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.va.1 $48$ $2$ $2$ $3$ $3$ $1^{2}$
48.96.3.vb.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.vc.1 $48$ $2$ $2$ $3$ $3$ $1^{2}$
48.96.3.vd.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ve.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.5.kl.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.kt.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.ut.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.vb.1 $48$ $2$ $2$ $5$ $5$ $1^{2}\cdot2$
120.240.17.frm.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.chcu.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.3.ebr.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ebs.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ebt.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ebu.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ebv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ebw.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.cdb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cdf.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cup.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cut.1 $240$ $2$ $2$ $5$ $?$ not computed