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Properties

Label 60.192.1-60.g.4.11
Level $60$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $3600$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.192.1.284

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}13&4\\36&11\end{bmatrix}$, $\begin{bmatrix}13&40\\54&1\end{bmatrix}$, $\begin{bmatrix}41&46\\0&47\end{bmatrix}$, $\begin{bmatrix}43&44\\24&23\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.96.1.g.4 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $11520$

Jacobian

Conductor: $2^{4}\cdot3^{2}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 3600.2.a.v

Models

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

$ 0 $ $=$ $ x^{2} + x y - x z - x w - y z - z w $
$=$ $6 x^{2} - x y + x z + x w - y^{2} + y z - 4 y w + 8 z^{2} + 3 z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} + 3 x^{3} y - 6 x^{3} z - 3 x^{2} y^{2} - 3 x^{2} y z + 7 x^{2} z^{2} - x y z^{2} - 8 x z^{3} + \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{1}{3^3\cdot5^6}\cdot\frac{116887905541617xz^{22}w-1154919060459675xz^{21}w^{2}+4917058352168697xz^{20}w^{3}-98176523761073940xz^{19}w^{4}+52545025823116176xz^{18}w^{5}-1586191026477145788xz^{17}w^{6}+2527445248550217954xz^{16}w^{7}-8884987268349642912xz^{15}w^{8}+18082065944578680765xz^{14}w^{9}-43329483509517235611xz^{13}w^{10}+51203751415097893101xz^{12}w^{11}-97462710463785026724xz^{11}w^{12}+130202986118672110455xz^{10}w^{13}-94392550217114270415xz^{9}w^{14}+43694892442331985015xz^{8}w^{15}-16158308230010730168xz^{7}w^{16}-117152594467181493885xz^{6}w^{17}+288778955406074734167xz^{5}w^{18}-301240177763429759085xz^{4}w^{19}+172552740584810446476xz^{3}w^{20}-56847274472056659633xz^{2}w^{21}+10160584169294024109xzw^{22}-765569373345618597xw^{23}+14442810021911y^{2}z^{22}-14312804760198y^{2}z^{21}w+1036260035780505y^{2}z^{20}w^{2}-732219430103640y^{2}z^{19}w^{3}+35280843428073960y^{2}z^{18}w^{4}-40170146394531420y^{2}z^{17}w^{5}+501930640278753273y^{2}z^{16}w^{6}-943451775975479616y^{2}z^{15}w^{7}+3750560762583147975y^{2}z^{14}w^{8}-7038305246476942470y^{2}z^{13}w^{9}+18670205716203543939y^{2}z^{12}w^{10}-30833486342931135240y^{2}z^{11}w^{11}+55451752438182317451y^{2}z^{10}w^{12}-85744211906111037930y^{2}z^{9}w^{13}+117461558124047505375y^{2}z^{8}w^{14}-135991759032872817984y^{2}z^{7}w^{15}+150152719680311198517y^{2}z^{6}w^{16}-141576696499323602970y^{2}z^{5}w^{17}+99620053090902352665y^{2}z^{4}w^{18}-47248082919849114360y^{2}z^{3}w^{19}+14164514669553947055y^{2}z^{2}w^{20}-2406075173371944162y^{2}zw^{21}+177331322602652589y^{2}w^{22}+28885620043822yz^{22}w-1991475542938287yz^{21}w^{2}+3498776776805832yz^{20}w^{3}-93888633179749380yz^{19}w^{4}+180075657574956690yz^{18}w^{5}-1658552420160632898yz^{17}w^{6}+3440673805192623396yz^{16}w^{7}-14312344684725744984yz^{15}w^{8}+28214488170344115990yz^{14}w^{9}-72470934167369265315yz^{13}w^{10}+134286391992312274176yz^{12}w^{11}-243507094072692951384yz^{11}w^{12}+380443378191474299832yz^{10}w^{13}-566578352576496442095yz^{9}w^{14}+712954196465670821340yz^{8}w^{15}-804669524968380002568yz^{7}w^{16}+825779617077790720998yz^{6}w^{17}-718378850470794173073yz^{5}w^{18}+467719728865610521680yz^{4}w^{19}-209646984228699569040yz^{3}w^{20}+60089611551987073032yz^{2}w^{21}-9886736699859505599yzw^{22}+707796057283923636yw^{23}+26357849671335z^{24}+97804165861353z^{23}w+1211372895575522z^{22}w^{2}-2257062625675536z^{21}w^{3}+22543800773002023z^{20}w^{4}+42521096928600114z^{19}w^{5}+725063885003224791z^{18}w^{6}+1094202526721961654z^{17}w^{7}+2654578194697459872z^{16}w^{8}-229794898984676739z^{15}w^{9}-1852634192203565064z^{14}w^{10}+8229590709495361956z^{13}w^{11}+11877136542438594804z^{12}w^{12}+33594167423838591729z^{11}w^{13}-31128617928664599084z^{10}w^{14}-27339582673396269090z^{9}w^{15}+30882197406307989348z^{8}w^{16}-17101350266958469593z^{7}w^{17}+134223974450606658384z^{6}w^{18}-244064306352336473952z^{5}w^{19}+203379180095091557034z^{4}w^{20}-93043902044694303999z^{3}w^{21}+24085871190691742394z^{2}w^{22}-3272176115176480938zw^{23}+176921565821318061w^{24}}{z^{12}(52920xz^{10}w-5514390xz^{9}w^{2}+9817710xz^{8}w^{3}-29949360xz^{7}w^{4}+20294589xz^{6}w^{5}-34748427xz^{5}w^{6}+24814125xz^{4}w^{7}+46384590xz^{3}w^{8}-85974945xz^{2}w^{9}+45248007xzw^{10}-7613991xw^{11}-432y^{2}z^{10}-6480y^{2}z^{9}w+3222315y^{2}z^{8}w^{2}-2721240y^{2}z^{7}w^{3}+13965345y^{2}z^{6}w^{4}-14731470y^{2}z^{5}w^{5}+26424160y^{2}z^{4}w^{6}-29327040y^{2}z^{3}w^{7}+26638365y^{2}z^{2}w^{8}-10877130y^{2}zw^{9}+1869327y^{2}w^{10}-864yz^{10}w-5898600yz^{9}w^{2}+11066760yz^{8}w^{3}-44181060yz^{7}w^{4}+66373380yz^{6}w^{5}-118275537yz^{5}w^{6}+138849080yz^{4}w^{7}-162259050yz^{3}w^{8}+117743760yz^{2}w^{9}-47036085yzw^{10}+7028028yw^{11}+6480z^{12}+44280z^{11}w+4211433z^{10}w^{2}+5409840z^{9}w^{3}+6371670z^{8}w^{4}+19628211z^{7}w^{5}+15720129z^{6}w^{6}+32056752z^{5}w^{7}-10077245z^{4}w^{8}-55941885z^{3}w^{9}+61532238z^{2}w^{10}-19697538zw^{11}+1748943w^{12})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.96.1.g.4 :

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

Equation of the image curve:

$0$ $=$ $ 3X^{4}+3X^{3}Y-3X^{2}Y^{2}-6X^{3}Z-3X^{2}YZ+7X^{2}Z^{2}-XYZ^{2}+Y^{2}Z^{2}-8XZ^{3}+YZ^{3}+4Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.96.0-12.a.2.9 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-12.a.2.15 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-60.a.2.5 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-60.a.2.27 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.1-60.c.1.6 $60$ $2$ $2$ $1$ $0$ dimension zero
60.96.1-60.c.1.10 $60$ $2$ $2$ $1$ $0$ 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
60.384.5-60.i.2.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.k.4.6 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.l.4.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.o.3.6 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.s.2.4 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.u.4.5 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.v.4.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.y.2.5 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.576.9-60.f.1.13 $60$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
60.960.33-60.k.3.13 $60$ $5$ $5$ $33$ $4$ $1^{16}\cdot8^{2}$
60.1152.33-60.k.1.25 $60$ $6$ $6$ $33$ $4$ $1^{16}\cdot8^{2}$
60.1920.65-60.k.2.19 $60$ $10$ $10$ $65$ $6$ $1^{32}\cdot8^{4}$
120.384.5-120.jg.4.10 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.jq.4.12 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.jy.4.10 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.kt.4.12 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.os.4.12 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.pc.4.10 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.pk.4.12 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.qf.4.10 $120$ $2$ $2$ $5$ $?$ not computed