↑

Properties

Label 60.288.9-60.fp.1.20
Level $60$
Index $288$
Genus $9$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $8$

Related objects

Downloads

Learn more

Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $360$
Index: $288$ $\PSL_2$-index:$144$
Genus: $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (all of which are rational) Cusp widths $2\cdot4\cdot6\cdot10\cdot12\cdot20\cdot30\cdot60$ Cusp orbits $1^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $8$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 60G9
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.288.9.1872

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&0\\24&37\end{bmatrix}$, $\begin{bmatrix}17&50\\18&17\end{bmatrix}$, $\begin{bmatrix}23&20\\24&19\end{bmatrix}$, $\begin{bmatrix}37&25\\24&31\end{bmatrix}$, $\begin{bmatrix}53&25\\0&53\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.9.fp.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $2$
Cyclic 60-torsion field degree: $32$
Full 60-torsion field degree: $7680$

Jacobian

Conductor: $2^{19}\cdot3^{15}\cdot5^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2^{3}$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 360.2.f.a, 360.2.f.c$^{2}$

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $ $=$ $ x v - x r + z t $
$=$ $w^{2} - w t + t u - t r - u s + r s$
$=$ $2 t^{2} + t u - t r + v r - r^{2}$
$=$ $w^{2} + w t + w u - t u + t r - u^{2} - u s + r^{2} + r s$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ - 15 x^{10} + 75 x^{9} y - 120 x^{8} y^{2} + 5 x^{7} y z^{2} + 210 x^{6} y^{4} - 20 x^{6} y^{2} z^{2} + \cdots + y^{8} z^{2} $
 Copy content Toggle raw display

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:1:0:0:-2:1)$, $(0:0:0:0:0:0:0:0:1)$, $(0:0:0:0:0:1:1:1:0)$, $(0:0:0:1:0:1:-1:0:1)$, $(0:0:0:-1/4:-1/4:1/2:1/4:1/2:1)$, $(0:0:0:0:0:-1:1:1:0)$, $(0:0:0:4/9:-1/9:-2/9:-4/9:-4/9:1)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2\,\frac{76796378657693274096wr^{9}-110575165275368948824wr^{8}s-515258010181853503700wr^{7}s^{2}+213066623746607006218wr^{6}s^{3}-328441153494494959846wr^{5}s^{4}+168784425027326548077wr^{4}s^{5}+979921851345669086050wr^{3}s^{6}-1880034637236788992626wr^{2}s^{7}+991216313789991105541wrs^{8}+3056066707272499200ws^{9}-1055806964898455949849tvs^{8}-69089671192917982560tr^{9}+217206775867060550688tr^{8}s+444264262106072301328tr^{7}s^{2}+970814774539527459968tr^{6}s^{3}-148759634697262840530tr^{5}s^{4}-867096187428263578050tr^{4}s^{5}-768911381488005260409tr^{3}s^{6}+1123656242682482419827tr^{2}s^{7}+2470870657256454096394trs^{8}-1339604397216206053274ts^{9}-62800156345932747280uvr^{8}+140059665488046055896uvr^{7}s-217673612686269437212uvr^{6}s^{2}+1254757387366251717910uvr^{5}s^{3}-11000919017669011784uvr^{4}s^{4}-1402979686353622287105uvr^{3}s^{5}+1275553240803934939627uvr^{2}s^{6}+869581768181423855828uvrs^{7}-1103564816213924880982uvs^{8}+68534377451750494736ur^{9}-253374653525120223512ur^{8}s+309293915133527348412ur^{7}s^{2}-434745021961052216246ur^{6}s^{3}-185116113862125974344ur^{5}s^{4}+1508771255116753699333ur^{4}s^{5}-1116457222892728107989ur^{3}s^{6}-1720925012711732812458ur^{2}s^{7}+3220112715249040187637urs^{8}-960325703972559000581us^{9}+524649450339303424v^{9}s-390280560942972928v^{8}s^{2}+7352445734024708096v^{7}s^{3}-730493441043472384v^{6}s^{4}+9418427145471148032v^{5}s^{5}+97068817497157910528v^{4}s^{6}+232483545674249502720v^{3}s^{7}+117159594161440826368v^{2}r^{8}-91659544865875019312v^{2}r^{7}s-304817549584285328232v^{2}r^{6}s^{2}-663511173345069253660v^{2}r^{5}s^{3}+402862601905968846766v^{2}r^{4}s^{4}+2008167390667132282398v^{2}r^{3}s^{5}+1207630045993903755129v^{2}r^{2}s^{6}-1800315513450322534922v^{2}rs^{7}-504232455477838356480v^{2}s^{8}-168678688799048705040vr^{9}-91619558412080391512vr^{8}s+663135603563022143052vr^{7}s^{2}+153917278845341599258vr^{6}s^{3}-99656912514190105422vr^{5}s^{4}-1395271994337850540015vr^{4}s^{5}-2282535646329243550246vr^{3}s^{6}+2594348005083619277868vr^{2}s^{7}+1094957112009758964724vrs^{8}-455332916573821497775vs^{9}+57380391325925212176r^{10}+216865637432563783016r^{9}s-257449687192411852772r^{8}s^{2}-86958327154493434062r^{7}s^{3}-12884733626732623984r^{6}s^{4}-441973845048884475509r^{5}s^{5}+602752348775672127317r^{4}s^{6}+14628712369784023127r^{3}s^{7}-1353280330843836201272r^{2}s^{8}+462835948437831635375rs^{9}+312399343209553920s^{10}}{4914394823580800wr^{9}-121061610264863360wr^{8}s+440364710423054880wr^{7}s^{2}+768726969997354848wr^{6}s^{3}-712295003029865872wr^{5}s^{4}-2390589947918012004wr^{4}s^{5}+549749640118119008wr^{3}s^{6}-611900164440277167wr^{2}s^{7}+1436108643087933407wrs^{8}-4391037389448310041tvs^{8}+6318300349510400tr^{9}+10734687691122560tr^{8}s-352399750231422400tr^{7}s^{2}-590290946586347360tr^{6}s^{3}+1768305595827560736tr^{5}s^{4}+5248598072429369208tr^{4}s^{5}+3939928939204335932tr^{3}s^{6}-1130693776188732218tr^{2}s^{7}+1027564961395449946trs^{8}-4292631406351662222ts^{9}+6655537605962880uvr^{8}-40688585754869760uvr^{7}s-186388946119037920uvr^{6}s^{2}+58647397090530880uvr^{5}s^{3}+701954089422885776uvr^{4}s^{4}+1183522095574613260uvr^{3}s^{5}-1225773344669635828uvr^{2}s^{6}-189434091071224883uvrs^{7}-647939377059047012uvs^{8}-6655537605962880ur^{9}+29134212715153920ur^{8}s+398206117608719840ur^{7}s^{2}-418183683850002752ur^{6}s^{3}-1566971435125760912ur^{5}s^{4}-173519505725883692ur^{4}s^{5}+4271937398361865668ur^{3}s^{6}+196627531554950499ur^{2}s^{7}+975595512243916068urs^{8}-1436108643087933407us^{9}-1049298900678606848v^{3}s^{7}+9299179486330880v^{2}r^{8}-59841139120323200v^{2}r^{7}s+229456583411452800v^{2}r^{6}s^{2}+1057538499845037024v^{2}r^{5}s^{3}+551663069982247264v^{2}r^{4}s^{4}-3205206349611405072v^{2}r^{3}s^{5}-3449808598454077356v^{2}r^{2}s^{6}-627993783355185576v^{2}rs^{7}+780561121885945856v^{2}s^{8}-19986705108763520vr^{9}+179098225892791680vr^{8}s-365712899024626400vr^{7}s^{2}-2359064099204249760vr^{6}s^{3}-1965120165553557200vr^{5}s^{4}+4491002809278461980vr^{4}s^{5}+5261423464349310400vr^{3}s^{6}+2660426089700209177vr^{2}s^{7}-2497238610693336272vrs^{8}+1041690756535666309vs^{9}+10687525622432640r^{10}-101384413383242240r^{9}s-66928385944684320r^{8}s^{2}+1307783153938720768r^{7}s^{3}+2208598950721117712r^{6}s^{4}-685471345597021004r^{5}s^{5}-2950167974245440684r^{4}s^{6}-1507511624344615365r^{3}s^{7}-98154517920672090r^{2}s^{8}-1041690756535666309rs^{9}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.9.fp.1 :

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

Equation of the image curve:

$0$ $=$ $ -15X^{10}+75X^{9}Y-120X^{8}Y^{2}+5X^{7}YZ^{2}+210X^{6}Y^{4}-20X^{6}Y^{2}Z^{2}-210X^{5}Y^{5}+24X^{5}Y^{3}Z^{2}-23X^{4}Y^{4}Z^{2}+120X^{3}Y^{7}-7X^{3}Y^{5}Z^{2}-75X^{2}Y^{8}+58X^{2}Y^{6}Z^{2}-3X^{2}Y^{4}Z^{4}+15XY^{9}-6XY^{7}Z^{2}+3XY^{5}Z^{4}+Y^{8}Z^{2} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
60.144.3-30.a.1.5 $60$ $2$ $2$ $3$ $0$ $2^{3}$
60.144.3-30.a.1.9 $60$ $2$ $2$ $3$ $0$ $2^{3}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.576.17-60.i.1.21 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.s.2.15 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.bk.2.1 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.bn.2.11 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.fd.1.3 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.fg.2.4 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{2}$
60.576.17-60.fh.2.1 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{2}$
60.576.17-60.fk.2.3 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.gm.1.11 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.gs.1.8 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{2}$
60.576.17-60.gu.1.2 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{2}$
60.576.17-60.gx.1.4 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{2}$
60.576.17-60.hk.1.7 $60$ $2$ $2$ $17$ $2$ $1^{4}\cdot2^{2}$
60.576.17-60.hn.1.4 $60$ $2$ $2$ $17$ $4$ $1^{4}\cdot2^{2}$
60.576.17-60.ho.1.2 $60$ $2$ $2$ $17$ $4$ $1^{4}\cdot2^{2}$
60.576.17-60.hr.1.4 $60$ $2$ $2$ $17$ $2$ $1^{4}\cdot2^{2}$
60.576.17-60.im.2.25 $60$ $2$ $2$ $17$ $3$ $1^{6}\cdot2$
60.576.17-60.ip.2.15 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2$
60.576.17-60.iy.2.9 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2$
60.576.17-60.jb.2.11 $60$ $2$ $2$ $17$ $4$ $1^{6}\cdot2$
60.576.17-60.jc.1.22 $60$ $2$ $2$ $17$ $2$ $1^{6}\cdot2$
60.576.17-60.jf.1.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2$
60.576.17-60.jo.1.6 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2$
60.576.17-60.jr.1.20 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2$
60.864.29-60.cce.2.8 $60$ $3$ $3$ $29$ $0$ $1^{8}\cdot2^{6}$
60.1440.49-60.bsg.1.11 $60$ $5$ $5$ $49$ $4$ $1^{16}\cdot2^{12}$