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Properties

Label 168.192.1-24.w.1.10
Level $168$
Index $192$
Genus $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $168$ $\SL_2$-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 $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 96$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1

Level structure

$\GL_2(\Z/168\Z)$-generators: $\begin{bmatrix}31&24\\0&83\end{bmatrix}$, $\begin{bmatrix}87&76\\152&99\end{bmatrix}$, $\begin{bmatrix}97&4\\40&167\end{bmatrix}$, $\begin{bmatrix}135&112\\64&89\end{bmatrix}$, $\begin{bmatrix}159&160\\92&141\end{bmatrix}$
Contains $-I$: no $\quad$ (see 24.96.1.w.1 for the level structure with $-I$)
Cyclic 168-isogeny field degree: $64$
Cyclic 168-torsion field degree: $1536$
Full 168-torsion field degree: $774144$

Jacobian

Conductor: $?$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

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

$ 0 $ $=$ $ 2 x w + y^{2} + 2 y z $
$=$ $3 x^{2} - y^{2} + 2 y z - 2 y w + 2 z^{2} - 4 z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} + 6 x^{3} y + 3 x^{2} y^{2} - 4 x^{2} z^{2} + 4 x y z^{2} - 8 x z^{3} + 2 y^{2} z^{2} + \cdots + 4 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 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{2^6}{3^2}\cdot\frac{41754382673351040xyz^{22}+649545757699829760xyz^{21}w+4621023206169822720xyz^{20}w^{2}+19645336613134310400xyz^{19}w^{3}+54516386907476867520xyz^{18}w^{4}+100280693034182734272xyz^{17}w^{5}+114888650016399663456xyz^{16}w^{6}+58463076469080849408xyz^{15}w^{7}-35411132795739860160xyz^{14}w^{8}-72644426174778750720xyz^{13}w^{9}-25249146899138464320xyz^{12}w^{10}+24715135295103732480xyz^{11}w^{11}+18142323540190901856xyz^{10}w^{12}-5075613180094039776xyz^{9}w^{13}-5277394354380944400xyz^{8}w^{14}+1178381769350711040xyz^{7}w^{15}+737689000423892472xyz^{6}w^{16}-233767276848173376xyz^{5}w^{17}-20855589151714992xyz^{4}w^{18}+14600436873144192xyz^{3}w^{19}-1441808987037156xyz^{2}w^{20}-31088255574612xyzw^{21}+6823986578214xyw^{22}-59049614472397056xz^{22}w-918596415820647168xz^{21}w^{2}-6511353999133764096xz^{20}w^{3}-27446687921770122240xz^{19}w^{4}-74960142542266782528xz^{18}w^{5}-133884629758502827776xz^{17}w^{6}-143992675008597581568xz^{16}w^{7}-56236724788619063808xz^{15}w^{8}+68804762127473796480xz^{14}w^{9}+98705817363404396160xz^{13}w^{10}+17359407165636686592xz^{12}w^{11}-43619889486553761024xz^{11}w^{12}-19478546964136054368xz^{10}w^{13}+12418498672806236544xz^{9}w^{14}+5850374139812668608xz^{8}w^{15}-3028213549077181056xz^{7}w^{16}-577155140442425232xz^{6}w^{17}+444217635263274192xz^{5}w^{18}-35004182335982496xz^{4}w^{19}-15131552741385984xz^{3}w^{20}+3125755911440028xz^{2}w^{21}-153323068201584xzw^{22}-2274663049568xw^{23}-59049614336348160yz^{23}-932514546507555456yz^{22}w-6727869218443977216yz^{21}w^{2}-28953427872736172160yz^{20}w^{3}-81033392556484212672yz^{19}w^{4}-149040396591840997632yz^{18}w^{5}-166285686133070613504yz^{17}w^{6}-68873815546907645088yz^{16}w^{7}+85193016292778266368yz^{15}w^{8}+134173258442561502144yz^{14}w^{9}+33164444216658713472yz^{13}w^{10}-61352577861605495424yz^{12}w^{11}-37729402261950093984yz^{11}w^{12}+16566085204355207808yz^{10}w^{13}+14167736406834427008yz^{9}w^{14}-4337979042676267152yz^{8}w^{15}-2606684227998436800yz^{7}w^{16}+1018946996910201816yz^{6}w^{17}+119747454397747200yz^{5}w^{18}-98123967333692568yz^{4}w^{19}+11336687775719316yz^{3}w^{20}+854718014980944yz^{2}w^{21}-220009654453024yzw^{22}+9098649627782yw^{23}+11337408z^{24}-27836255387665152z^{23}w-357674787402324480z^{22}w^{2}-1975624998733393920z^{21}w^{3}-5762984126494759488z^{20}w^{4}-7706335791035937024z^{19}w^{5}+4307005262379018240z^{18}w^{6}+34922797470528307776z^{17}w^{7}+55526433921901263600z^{16}w^{8}+26494400963409114240z^{15}w^{9}-30604125281363775360z^{14}w^{10}-41605073813002944384z^{13}w^{11}+925945397538177312z^{12}w^{12}+22567097454078409344z^{11}w^{13}+2789212983171167808z^{10}w^{14}-7902784453789675872z^{9}w^{15}-136913848856804844z^{8}w^{16}+1693381336122763728z^{7}w^{17}-320719360560380352z^{6}w^{18}-112155562586182752z^{5}w^{19}+47679871510658412z^{4}w^{20}-4620999804863760z^{3}w^{21}-365518638626864z^{2}w^{22}+87580870540804zw^{23}-3791104093231w^{24}}{w^{4}(22674816xyz^{17}w-471132288xyz^{16}w^{2}+4625662464xyz^{15}w^{3}-14347634690880xyz^{14}w^{4}-141626719757376xyz^{13}w^{5}-582789021561072xyz^{12}w^{6}-1206161607398112xyz^{11}w^{7}-1096207686424584xyz^{10}w^{8}+233928055539432xyz^{9}w^{9}+1085751957405636xyz^{8}w^{10}+177242293729152xyz^{7}w^{11}-568777772147490xyz^{6}w^{12}+82363283781408xyz^{5}w^{13}+77813312604192xyz^{4}w^{14}-29077104741156xyz^{3}w^{15}+2731842868446xyz^{2}w^{16}+132721391790xyzw^{17}-22075240722xyw^{18}+47869056xz^{18}w-997691904xz^{17}w^{2}+9786702528xz^{16}w^{3}-60042912768xz^{15}w^{4}+20508207518016xz^{14}w^{5}+199644163435584xz^{13}w^{6}+817467764871888xz^{12}w^{7}+1634331341425536xz^{11}w^{8}+1324771881248232xz^{10}w^{9}-652194368573184xz^{9}w^{10}-1584150596849520xz^{8}w^{11}+74486223588384xz^{7}w^{12}+895928783759820xz^{6}w^{13}-332438389601724xz^{5}w^{14}-38605367136906xz^{4}w^{15}+40925526865992xz^{3}w^{16}-7459442158590xz^{2}w^{17}+385755031980xzw^{18}+7358415134xw^{19}-2519424yz^{19}+55427328yz^{18}w-512702784yz^{17}w^{2}+2410668864yz^{16}w^{3}+20246651835840yz^{15}w^{4}+205210396372896yz^{14}w^{5}+862965196641360yz^{13}w^{6}+1819593712916064yz^{12}w^{7}+1627508671523640yz^{11}w^{8}-610663706770032yz^{10}w^{9}-2083642737108912yz^{9}w^{10}-344092435824060yz^{8}w^{11}+1358695169640000yz^{7}w^{12}-96206850062082yz^{6}w^{13}-367717453396254yz^{5}w^{14}+143469288414504yz^{4}w^{15}-10984709848878yz^{3}w^{16}-3365107977978yz^{2}w^{17}+655070100460yzw^{18}-29433655856yw^{19}-5038848z^{20}+110854656z^{19}w-1068235776z^{18}w^{2}+5708174976z^{17}w^{3}-15819358320z^{16}w^{4}+9541018797120z^{15}w^{5}+68930676694656z^{14}w^{6}+153993992799168z^{13}w^{7}-26495428280040z^{12}w^{8}-661847559342912z^{11}w^{9}-701273948816448z^{10}w^{10}+358266293575992z^{9}w^{11}+939479059458981z^{8}w^{12}-492818764418748z^{7}w^{13}-248960269770372z^{6}w^{14}+254073059391780z^{5}w^{15}-68772903348888z^{4}w^{16}+3527216713980z^{3}w^{17}+1538296644314z^{2}w^{18}-268008822436zw^{19}+12264023422w^{20})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.w.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 2z$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ 3X^{4}+6X^{3}Y+3X^{2}Y^{2}-4X^{2}Z^{2}+4XYZ^{2}+2Y^{2}Z^{2}-8XZ^{3}-8YZ^{3}+4Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.96.0-8.c.1.7 $56$ $2$ $2$ $0$ $0$ full Jacobian
168.96.0-24.b.1.12 $168$ $2$ $2$ $0$ $?$ full Jacobian
168.96.0-24.b.1.22 $168$ $2$ $2$ $0$ $?$ full Jacobian
168.96.0-8.c.1.4 $168$ $2$ $2$ $0$ $?$ full Jacobian
168.96.1-24.n.2.19 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1-24.n.2.20 $168$ $2$ $2$ $1$ $?$ 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
168.384.5-24.be.1.5 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.be.1.8 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bg.1.3 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bg.1.5 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bh.2.1 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bh.2.6 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bj.1.6 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-24.bj.1.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hl.2.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hl.2.14 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hm.1.7 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hm.1.9 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hq.2.5 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hq.2.11 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hs.1.8 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.hs.1.11 $168$ $2$ $2$ $5$ $?$ not computed