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Properties

Label 60.96.1.b.1
Level $60$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $24$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ 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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.377

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&20\\42&53\end{bmatrix}$, $\begin{bmatrix}29&0\\6&11\end{bmatrix}$, $\begin{bmatrix}37&4\\6&23\end{bmatrix}$, $\begin{bmatrix}53&10\\54&53\end{bmatrix}$, $\begin{bmatrix}53&16\\24&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.192.1-60.b.1.1, 60.192.1-60.b.1.2, 60.192.1-60.b.1.3, 60.192.1-60.b.1.4, 60.192.1-60.b.1.5, 60.192.1-60.b.1.6, 60.192.1-60.b.1.7, 60.192.1-60.b.1.8, 60.192.1-60.b.1.9, 60.192.1-60.b.1.10, 60.192.1-60.b.1.11, 60.192.1-60.b.1.12, 120.192.1-60.b.1.1, 120.192.1-60.b.1.2, 120.192.1-60.b.1.3, 120.192.1-60.b.1.4, 120.192.1-60.b.1.5, 120.192.1-60.b.1.6, 120.192.1-60.b.1.7, 120.192.1-60.b.1.8, 120.192.1-60.b.1.9, 120.192.1-60.b.1.10, 120.192.1-60.b.1.11, 120.192.1-60.b.1.12, 120.192.1-60.b.1.13, 120.192.1-60.b.1.14, 120.192.1-60.b.1.15, 120.192.1-60.b.1.16, 120.192.1-60.b.1.17, 120.192.1-60.b.1.18, 120.192.1-60.b.1.19, 120.192.1-60.b.1.20, 120.192.1-60.b.1.21, 120.192.1-60.b.1.22, 120.192.1-60.b.1.23, 120.192.1-60.b.1.24, 120.192.1-60.b.1.25, 120.192.1-60.b.1.26, 120.192.1-60.b.1.27, 120.192.1-60.b.1.28, 120.192.1-60.b.1.29, 120.192.1-60.b.1.30, 120.192.1-60.b.1.31, 120.192.1-60.b.1.32, 120.192.1-60.b.1.33, 120.192.1-60.b.1.34, 120.192.1-60.b.1.35, 120.192.1-60.b.1.36, 120.192.1-60.b.1.37, 120.192.1-60.b.1.38, 120.192.1-60.b.1.39, 120.192.1-60.b.1.40, 120.192.1-60.b.1.41, 120.192.1-60.b.1.42, 120.192.1-60.b.1.43, 120.192.1-60.b.1.44
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

$ 0 $ $=$ $ 3 x^{2} - 5 x y + 2 x z + 2 z^{2} $
$=$ $3 x^{2} + 5 x y + 2 x z - 5 y^{2} + 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 175 x^{4} - 10 x^{3} y - 2 x^{2} y^{2} + 20 x^{2} z^{2} - 2 x y z^{2} - 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10z$
$\displaystyle Z$ $=$ $\displaystyle w$

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}{3^{12}\cdot5^2\cdot7^4}\cdot\frac{115123898811316366505195200000000000xz^{23}-275393598143168346333275640000000000xz^{21}w^{2}+161706757926027269613150360000000000xz^{19}w^{4}-140913457983221560800488892000000000xz^{17}w^{6}+33721006297883428805022383040000000xz^{15}w^{8}-27677573190680910630799015224000000xz^{13}w^{10}+4267523032924807604337210763200000xz^{11}w^{12}-2784300522778997291422790287200000xz^{9}w^{14}+305878760975452447439578298160000xz^{7}w^{16}-59947104293947185750497087348400xz^{5}w^{18}+3040700449170839976093581109360xz^{3}w^{20}-94889555612958011306487113160xzw^{22}-842135056105990361826718400000000000y^{2}z^{22}+392110749842916311682674220000000000y^{2}z^{20}w^{2}-802324761831906379366194720000000000y^{2}z^{18}w^{4}+206324664078323572727011794000000000y^{2}z^{16}w^{6}-220265435324505366996383685600000000y^{2}z^{14}w^{8}+37351932135268637674026396108000000y^{2}z^{12}w^{10}-23902588252381887415743141452400000y^{2}z^{10}w^{12}+3042380602673856616013332646400000y^{2}z^{8}w^{14}-684818310317886495312832244016000y^{2}z^{6}w^{16}+65710307242263632018853703851000y^{2}z^{4}w^{18}-6111989026533759445530816945240y^{2}z^{2}w^{20}+256911915558171431750321548188y^{2}w^{22}-569594434029975610664690400000000000yz^{23}+178938721093376548127451600000000000yz^{21}w^{2}-527950631726815626217899360000000000yz^{19}w^{4}+153572580594461236884254208000000000yz^{17}w^{6}-174392755228449925697721673440000000yz^{15}w^{8}+42231107107666030546645249968000000yz^{13}w^{10}-21990824796298874889646508256000000yz^{11}w^{12}+4029546752457935259743776483200000yz^{9}w^{14}-721534548205089403122535219512000yz^{7}w^{16}+67843607833363384547990303479200yz^{5}w^{18}-3312316675052683960743763799040yz^{3}w^{20}+147119528380290390905381800000000000z^{24}-316018399546954446370896520000000000z^{22}w^{2}+219198948707142964127665188000000000z^{20}w^{4}-156321483339295683956908620000000000z^{18}w^{6}+39925007215090470439129453500000000z^{16}w^{8}-23123739829277286061669668456000000z^{14}w^{10}+2114167771106681314006286236800000z^{12}w^{12}-1708960385811474056272933409040000z^{10}w^{14}+16298911741241648860604598510000z^{8}w^{16}-17580288771419382274674719077200z^{6}w^{18}-2834779682236611531637105185960z^{4}w^{20}+338870976606632079959231397312z^{2}w^{22}-22234213320029093419337858067w^{24}}{w^{4}(6513448955094272250000xz^{17}w^{2}-1312058989480938300000xz^{15}w^{4}-10991499076540867500xz^{13}w^{6}+6896672073858689500xz^{11}w^{8}+3110955868469904000xz^{9}w^{10}-638049908378185380xz^{7}w^{12}+49217277202498188xz^{5}w^{14}-1777432910688540xz^{3}w^{16}+25215239574000xzw^{18}+52906109997713523750000y^{2}z^{18}-11223776775429847125000y^{2}z^{16}w^{2}+197927273476044000000y^{2}z^{14}w^{4}+36730003973118198750y^{2}z^{12}w^{6}+10046253877540066375y^{2}z^{10}w^{8}-681273764343822825y^{2}z^{8}w^{10}-183243076619112405y^{2}z^{6}w^{12}+25241267526983550y^{2}z^{4}w^{14}-1180808656550775y^{2}z^{2}w^{16}+19611853002000y^{2}w^{18}+43422993033961815000000yz^{19}-13226622362742978000000yz^{17}w^{2}+1196369341797227850000yz^{15}w^{4}-8722468774650645000yz^{13}w^{6}+2127147900854655000yz^{11}w^{8}-1362298026669249600yz^{9}w^{10}+132038379957322440yz^{7}w^{12}-5355850299483600yz^{5}w^{14}+81809443951200yz^{3}w^{16}+4342299303396181500000z^{18}w^{2}-406786790776302956250z^{16}w^{4}-96886166233854532500z^{14}w^{6}+2755926813381868000z^{12}w^{8}+1954068486731237975z^{10}w^{10}+82774646084923695z^{8}w^{12}-60050125828257201z^{6}w^{14}+6044451587099730z^{4}w^{16}-251851213711755z^{2}w^{18}+3922370600400w^{20})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.1.b.1 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0.a.2 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0.c.4 $60$ $2$ $2$ $0$ $0$ 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
60.192.5.b.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.e.3 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.192.5.r.4 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.u.4 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.288.9.b.1 $60$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
60.480.33.f.2 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot8^{2}$
60.576.33.f.3 $60$ $6$ $6$ $33$ $0$ $1^{16}\cdot8^{2}$
60.960.65.f.3 $60$ $10$ $10$ $65$ $2$ $1^{32}\cdot8^{4}$
120.192.5.f.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ih.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ma.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mt.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mw.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nd.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ne.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nj.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nm.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ns.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nt.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.nu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ok.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pb.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.qx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.9.cb.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.co.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.fh.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.fl.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.lp.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.lu.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.mr.3 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.mw.3 $120$ $2$ $2$ $9$ $?$ not computed
180.288.9.b.4 $180$ $3$ $3$ $9$ $?$ not computed
180.288.17.b.4 $180$ $3$ $3$ $17$ $?$ not computed
180.288.17.j.3 $180$ $3$ $3$ $17$ $?$ not computed