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Properties

Label 8.96.1.l.2
Level $8$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
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 $2^{8}$
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 and Zureick-Brown (RZB) label: X445
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.180

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&7\\0&3\end{bmatrix}$, $\begin{bmatrix}7&0\\0&3\end{bmatrix}$, $\begin{bmatrix}7&5\\0&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.l.2.1, 8.192.1-8.l.2.2, 16.192.1-8.l.2.1, 16.192.1-8.l.2.2, 24.192.1-8.l.2.1, 24.192.1-8.l.2.2, 40.192.1-8.l.2.1, 40.192.1-8.l.2.2, 48.192.1-8.l.2.1, 48.192.1-8.l.2.2, 56.192.1-8.l.2.1, 56.192.1-8.l.2.2, 80.192.1-8.l.2.1, 80.192.1-8.l.2.2, 88.192.1-8.l.2.1, 88.192.1-8.l.2.2, 104.192.1-8.l.2.1, 104.192.1-8.l.2.2, 112.192.1-8.l.2.1, 112.192.1-8.l.2.2, 120.192.1-8.l.2.1, 120.192.1-8.l.2.2, 136.192.1-8.l.2.1, 136.192.1-8.l.2.2, 152.192.1-8.l.2.1, 152.192.1-8.l.2.2, 168.192.1-8.l.2.1, 168.192.1-8.l.2.2, 176.192.1-8.l.2.1, 176.192.1-8.l.2.2, 184.192.1-8.l.2.1, 184.192.1-8.l.2.2, 208.192.1-8.l.2.1, 208.192.1-8.l.2.2, 232.192.1-8.l.2.1, 232.192.1-8.l.2.2, 240.192.1-8.l.2.1, 240.192.1-8.l.2.2, 248.192.1-8.l.2.1, 248.192.1-8.l.2.2, 264.192.1-8.l.2.1, 264.192.1-8.l.2.2, 272.192.1-8.l.2.1, 272.192.1-8.l.2.2, 280.192.1-8.l.2.1, 280.192.1-8.l.2.2, 296.192.1-8.l.2.1, 296.192.1-8.l.2.2, 304.192.1-8.l.2.1, 304.192.1-8.l.2.2, 312.192.1-8.l.2.1, 312.192.1-8.l.2.2, 328.192.1-8.l.2.1, 328.192.1-8.l.2.2
Cyclic 8-isogeny field degree: $1$
Cyclic 8-torsion field degree: $2$
Full 8-torsion field degree: $16$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ x^{2} + x y + x z + x w + y z + 2 y w + z w $
$=$ $x^{2} - x y - 5 x z - x w + y^{2} + y z + z^{2} + z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 7 x^{4} + 44 x^{3} y + 44 x^{3} z + 34 x^{2} y^{2} + 108 x^{2} y z + 34 x^{2} z^{2} + 12 x y^{3} + \cdots + 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

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{2^2}{3^4}\cdot\frac{40335232082679986400xz^{22}w+887836592586868251840xz^{21}w^{2}+9254380280865479514432xz^{20}w^{3}+60902617877428756515840xz^{19}w^{4}+284788387373224742368128xz^{18}w^{5}+1009976836114414935601920xz^{17}w^{6}+2831008306685276079537408xz^{16}w^{7}+6450014210861724123709440xz^{15}w^{8}+12176504071968062256233472xz^{14}w^{9}+19292488517105203766777856xz^{13}w^{10}+25853056626639897247279104xz^{12}w^{11}+29403265214094446635941888xz^{11}w^{12}+28371814521423770471804928xz^{10}w^{13}+23134916159902353796816896xz^{9}w^{14}+15820843266087648655122432xz^{8}w^{15}+8968596515958278456082432xz^{7}w^{16}+4146433280568927177891840xz^{6}w^{17}+1529191597082478990704640xz^{5}w^{18}+436462813832167203815424xz^{4}w^{19}+92345161098326365175808xz^{3}w^{20}+13547089960269717274624xz^{2}w^{21}+1222044281748601110528xzw^{22}+50705500426296426496xw^{23}-22279740805235116476y^{2}z^{22}-514355436964780554312y^{2}z^{21}w-5644999123201646164944y^{2}z^{20}w^{2}-39178517130021163969440y^{2}z^{19}w^{3}-193054637581257127522032y^{2}z^{18}w^{4}-719093465271256797601632y^{2}z^{17}w^{5}-2105205854800226677121088y^{2}z^{16}w^{6}-4971961087147987752577536y^{2}z^{15}w^{7}-9643939557185886007374720y^{2}z^{14}w^{8}-15549035871088787247924480y^{2}z^{13}w^{9}-20994038439589575691688448y^{2}z^{12}w^{10}-23819228400989056778431488y^{2}z^{11}w^{11}-22702271161732976856024576y^{2}z^{10}w^{12}-18101507009809691487144960y^{2}z^{9}w^{13}-11972573569114004499572736y^{2}z^{8}w^{14}-6479719624476653021503488y^{2}z^{7}w^{15}-2811906799025424780266496y^{2}z^{6}w^{16}-950195874090198753994752y^{2}z^{5}w^{17}-239637944216575814307840y^{2}z^{4}w^{18}-42257593821879347355648y^{2}z^{3}w^{19}-4648530139042511499264y^{2}z^{2}w^{20}-245775451609108439040y^{2}zw^{21}-1293462322347851776y^{2}w^{22}-44559481610470232952yz^{23}-1009172589090208421184yz^{22}w-10835226835072191075456yz^{21}w^{2}-73365199472477046256272yz^{20}w^{3}-351594169057275027271200yz^{19}w^{4}-1268739635409839852292096yz^{18}w^{5}-3579168105368812346285952yz^{17}w^{6}-8082409647436711154383680yz^{16}w^{7}-14816203275965662428297984yz^{15}w^{8}-22176103599287305786312704yz^{14}w^{9}-27015082583767819170490368yz^{13}w^{10}-26350914509178057998553600yz^{12}w^{11}-19695354039642902783394816yz^{11}w^{12}-9856938249406089005727744yz^{10}w^{13}-1141305453326952234160128yz^{9}w^{14}+3577185042788310245701632yz^{8}w^{15}+4303607717815312272500736yz^{7}w^{16}+2952101806986864952590336yz^{6}w^{17}+1408921239912519022018560yz^{5}w^{18}+482685807013832719577088yz^{4}w^{19}+116625432330450015117312yz^{3}w^{20}+18826371923081191489536yz^{2}w^{21}+1813219414778181877760yzw^{22}+78230670576040263680yw^{23}-7426300779465882381z^{24}-204129625243228838352z^{23}w-2570700229688381303556z^{22}w^{2}-19947638653098090715512z^{21}w^{3}-107830507735191164283720z^{20}w^{4}-434467544577449778829728z^{19}w^{5}-1361567914218057432840336z^{18}w^{6}-3415779969041914022215584z^{17}w^{7}-6997933936841034728197296z^{16}w^{8}-11868298172235210506158080z^{15}w^{9}-16809218519049240875256960z^{14}w^{10}-19979460364572070026400512z^{13}w^{11}-19966327439902701927724800z^{12}w^{12}-16767623370906820672035840z^{11}w^{13}-11812367329849901827754496z^{10}w^{14}-6972925378204501146031104z^{9}w^{15}-3459553423705779902952192z^{8}w^{16}-1459849990410225521160192z^{7}w^{17}-535140631542216034464768z^{6}w^{18}-172897511763347199154176z^{5}w^{19}-48104250289637917575168z^{4}w^{20}-10696420570368940679168z^{3}w^{21}-1697476538376446332928z^{2}w^{22}-164024223306321567744zw^{23}-7015158961969909760w^{24}}{(z+w)^{4}(4649119679250000xz^{18}w+96425345125980000xz^{17}w^{2}+922613395280608800xz^{16}w^{3}+5390829511728791040xz^{15}w^{4}+21440945933247162816xz^{14}w^{5}+61206922271880206208xz^{13}w^{6}+128766063920571888768xz^{12}w^{7}+201633751855510179840xz^{11}w^{8}+234214836761347068672xz^{10}w^{9}+198066960981527367168xz^{9}w^{10}+116347799512444396032xz^{8}w^{11}+41378367822760894464xz^{7}w^{12}+3139309439210787840xz^{6}w^{13}-5379711660838066176xz^{5}w^{14}-3260314004394190848xz^{4}w^{15}-969919420977512448xz^{3}w^{16}-166000623831138304xz^{2}w^{17}-15368487822213120xzw^{18}-582071480074240xw^{19}-817213556250000y^{2}z^{18}-17499315820050000y^{2}z^{17}w-174623839374359250y^{2}z^{16}w^{2}-1066851607008466080y^{2}z^{15}w^{3}-4413939319406861784y^{2}z^{14}w^{4}-12915501723347844816y^{2}z^{13}w^{5}-27027356965226415360y^{2}z^{12}w^{6}-39618068472546769728y^{2}z^{11}w^{7}-37341980895975240096y^{2}z^{10}w^{8}-14837721734985067584y^{2}z^{9}w^{9}+13812628850039881152y^{2}z^{8}w^{10}+27885735558730828800y^{2}z^{7}w^{11}+23069103385947106176y^{2}z^{6}w^{12}+11466302693103002880y^{2}z^{5}w^{13}+3591310862802004992y^{2}z^{4}w^{14}+679938472191015936y^{2}z^{3}w^{15}+66283551424897536y^{2}z^{2}w^{16}+1377873294259200y^{2}zw^{17}-174273452415488y^{2}w^{18}-1634427112500000yz^{19}-31112319244725000yz^{18}w-265529036560108500yz^{17}w^{2}-1302229453328363160yz^{16}w^{3}-3779137925837942832yz^{15}w^{4}-4907117981567373696yz^{14}w^{5}+8447307710127746112yz^{13}w^{6}+59154344654409500256yz^{12}w^{7}+155151020669660728128yz^{11}w^{8}+256414710876657373440yz^{10}w^{9}+290874552120624198528yz^{9}w^{10}+228898318333466837376yz^{8}w^{11}+121101882210539926272yz^{7}w^{12}+38596746550473934848yz^{6}w^{13}+3983310555619875840yz^{5}w^{14}-2230744378608041472yz^{4}w^{15}-1122539524155823104yz^{3}w^{16}-232596782463928320yz^{2}w^{17}-23652579728221184yzw^{18}-936295734370304yw^{19}-272394267187500z^{20}-7031704018725000z^{19}w-78940427435192250z^{18}w^{2}-514663732997600760z^{17}w^{3}-2160373686145360245z^{16}w^{4}-6005204816361178752z^{15}w^{5}-10517756815805076936z^{14}w^{6}-8234857359403191216z^{13}w^{7}+10499865507553998672z^{12}w^{8}+43382071935238189248z^{11}w^{9}+69547578576079237344z^{10}w^{10}+69129111634629473088z^{9}w^{11}+46565781467703237792z^{8}w^{12}+22166021362346528256z^{7}w^{13}+7889647727282131584z^{6}w^{14}+2327421677380633344z^{5}w^{15}+621527566806453504z^{4}w^{16}+139538890743649280z^{3}w^{17}+21438394238796800z^{2}w^{18}+1753105119765504zw^{19}+46712709251840w^{20})}$

Modular covers

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Cover information

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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
8.48.0.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.1.bb.1 $8$ $2$ $2$ $1$ $0$ 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
16.192.5.bu.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.ca.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.17.ckn.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.rl.1 $24$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
40.480.33.wr.1 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.bjv.1 $40$ $6$ $6$ $33$ $1$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.cbj.1 $40$ $10$ $10$ $65$ $11$ $1^{28}\cdot2^{10}\cdot4^{4}$
48.192.5.gv.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hi.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
56.768.49.rl.2 $56$ $8$ $8$ $49$ $3$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.ckr.1 $56$ $21$ $21$ $145$ $19$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.cjn.2 $56$ $28$ $28$ $193$ $22$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
80.192.5.lt.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.mk.2 $80$ $2$ $2$ $5$ $?$ not computed
112.192.5.gv.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.hi.2 $112$ $2$ $2$ $5$ $?$ not computed
176.192.5.gv.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.hi.2 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.lt.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.mk.2 $208$ $2$ $2$ $5$ $?$ not computed
240.192.5.btt.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvm.2 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.lt.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.mk.2 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.gv.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.hi.2 $304$ $2$ $2$ $5$ $?$ not computed