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Properties

Label 60.288.7-60.me.1.11
Level $60$
Index $288$
Genus $7$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $8$

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Invariants

Level: $60$ $\SL_2$-level: $60$ Newform level: $60$
Index: $288$ $\PSL_2$-index:$144$
Genus: $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $8$ are rational) Cusp widths $1^{2}\cdot3^{2}\cdot4\cdot5^{2}\cdot12\cdot15^{2}\cdot20\cdot60$ Cusp orbits $1^{8}\cdot2^{2}$
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: 60O7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.288.7.2990

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&45\\52&11\end{bmatrix}$, $\begin{bmatrix}11&30\\28&49\end{bmatrix}$, $\begin{bmatrix}31&0\\18&53\end{bmatrix}$, $\begin{bmatrix}41&30\\12&11\end{bmatrix}$, $\begin{bmatrix}47&15\\40&59\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.144.7.me.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $2$
Cyclic 60-torsion field degree: $16$
Full 60-torsion field degree: $7680$

Jacobian

Conductor: $2^{7}\cdot3^{7}\cdot5^{7}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2^{2}$
Newforms: 15.2.a.a$^{2}$, 30.2.a.a, 30.2.c.a, 60.2.d.a

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ x^{2} - w t $
$=$ $x^{2} - z u$
$=$ $x t - x u + t v$
$=$ $x^{2} + x v - w u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{6} y - x^{5} z^{2} - 3 x^{4} y z^{2} - 3 x^{2} y^{3} z^{2} + x y^{4} z^{2} + 5 y^{3} z^{4} $
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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
$(0:0:1:0:0:0:0)$, $(-1:1:1:1:1:1:0)$, $(1:-1:1:1:1:1:0)$, $(0:0:0:0:1:0:0)$, $(0:0:0:1:0:0:0)$, $(0:0:0:0:0:1:0)$, $(-1/2:-1/2:-1/2:1/2:1/2:-1/2:1)$, $(-1/2:-1/2:1/2:-1/2:-1/2:1/2:1)$

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 \frac{1}{5}\cdot\frac{30277434060031250xv^{11}-7294921875yz^{10}v+1848902343750yz^{8}v^{3}-249820197656250yz^{6}v^{5}+1383027635390625yz^{4}v^{7}-22560653255859375yz^{2}v^{9}-2633522373610690ywu^{9}v-75786323433840640ywu^{7}v^{3}-490561846510011140ywu^{5}v^{5}-742983199064384525ywu^{3}v^{7}+1960823697962102845ywuv^{9}-961083984375yt^{10}v+19862402343750yt^{8}v^{3}-271452832031250yt^{6}v^{5}-1498071972655407yt^{4}v^{7}-588676500776610569yt^{2}v^{9}+154532744704440yu^{10}v-21373029232588997yu^{8}v^{3}-397905099953278182yu^{6}v^{5}-2208938851815795718yu^{4}v^{7}-5302508522410737670yu^{2}v^{9}-140062462655756250yv^{11}-48828125z^{12}+292968750z^{11}w+28378906250z^{10}v^{2}-1837080078125z^{9}wv^{2}-34558154296875z^{8}v^{4}+81811927734375z^{7}wv^{4}+74612048828125z^{6}v^{6}-3878989435546875z^{5}wv^{6}-7695830262265625z^{4}v^{8}-6093958864375000z^{3}wv^{8}-69235983566437500z^{2}v^{10}-344708395405593750zwv^{10}-35595703125w^{12}+42714843750w^{10}v^{2}-29900390625w^{8}v^{4}+19364062500w^{6}v^{6}-9568125000w^{4}v^{8}+2255343750w^{2}v^{10}-182205810546032wu^{11}-17909591420835390wu^{9}v^{2}-230211974028744200wu^{7}v^{4}-910334503965789340wu^{5}v^{6}-1245075152363231555wu^{3}v^{8}+25060799103556250wuv^{10}-35595703125t^{12}+213574218750t^{11}u+925488281250t^{10}v^{2}-7012353515625t^{9}uv^{2}-17619873046875t^{8}v^{4}+60761865234375t^{7}uv^{4}-248991943359375t^{6}v^{6}-1753978271484375t^{5}uv^{6}-21882657935619721t^{4}v^{8}-140357975514158784t^{3}uv^{8}-1982173492438153422t^{2}v^{10}+36263232421032tu^{11}+3169867280775048tu^{9}v^{2}+43413777186414069tu^{7}v^{4}+222424778124919235tu^{5}v^{6}+447480847073030397tu^{3}v^{8}+2150458570733668943tuv^{10}-48828125u^{12}+36263525389782u^{10}v^{2}+3023923677259423u^{8}v^{4}+28938268457074635u^{6}v^{6}+26524550882502937u^{4}v^{8}-462229524379339319u^{2}v^{10}-68941680242962500v^{12}}{v(3069832500xv^{10}-1953125yz^{10}-19921875yz^{8}v^{2}+15156250yz^{6}v^{4}-80796875yz^{4}v^{6}-805365625yz^{2}v^{8}+2441918ywu^{9}-193546665ywu^{7}v^{2}-8130736545ywu^{5}v^{4}-41959930695ywu^{3}v^{6}-19618058655ywuv^{8}-5960067644yt^{2}v^{8}-488793yu^{10}+50870372yu^{8}v^{2}-2027414626yu^{6}v^{4}-39466421591yu^{4}v^{6}-141593453635yu^{2}v^{8}-9147745000yv^{10}+7812500z^{10}v+23437500z^{9}wv-72656250z^{8}v^{3}-60937500z^{7}wv^{3}+52265625z^{6}v^{5}-44609375z^{5}wv^{5}-490906250z^{4}v^{7}-1667515625z^{3}wv^{7}-4790181250z^{2}v^{9}-17263468750zwv^{9}+6099168wu^{9}v-1829544895wu^{7}v^{3}-22721690650wu^{5}v^{5}-58881831895wu^{3}v^{7}-511208125wuv^{9}+1217379t^{4}v^{7}-355208874t^{3}uv^{7}-29264113932t^{2}v^{9}-728586tu^{9}v+323087980tu^{7}v^{3}+4327308392tu^{5}v^{5}+19210422640tu^{3}v^{7}+58161279683tuv^{9}-728586u^{8}v^{3}+327969769u^{6}v^{5}+2866591060u^{4}v^{7}+4569606u^{2}v^{9}-3453149375v^{11})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.7.me.1 :

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

Equation of the image curve:

$0$ $=$ $ X^{6}Y-X^{5}Z^{2}-3X^{4}YZ^{2}-3X^{2}Y^{3}Z^{2}+XY^{4}Z^{2}+5Y^{3}Z^{4} $

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.1.9 $60$ $2$ $2$ $3$ $0$ $2^{2}$
60.144.3-30.a.1.41 $60$ $2$ $2$ $3$ $0$ $2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.576.13-60.id.2.9 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2$
60.576.13-60.ie.2.43 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2$
60.576.13-60.mi.2.1 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2$
60.576.13-60.mk.2.9 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2$
60.576.13-60.nj.1.11 $60$ $2$ $2$ $13$ $1$ $1^{4}\cdot2$
60.576.13-60.nk.1.28 $60$ $2$ $2$ $13$ $0$ $1^{4}\cdot2$
60.576.13-60.oa.1.3 $60$ $2$ $2$ $13$ $4$ $1^{4}\cdot2$
60.576.13-60.oc.1.10 $60$ $2$ $2$ $13$ $2$ $1^{4}\cdot2$
60.576.17-60.k.2.29 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{3}$
60.576.17-60.q.2.18 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{3}$
60.576.17-60.fd.2.7 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{3}$
60.576.17-60.ff.2.2 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{3}$
60.576.17-60.go.1.11 $60$ $2$ $2$ $17$ $0$ $1^{4}\cdot2^{3}$
60.576.17-60.gq.1.11 $60$ $2$ $2$ $17$ $1$ $1^{4}\cdot2^{3}$
60.576.17-60.hk.1.7 $60$ $2$ $2$ $17$ $2$ $1^{4}\cdot2^{3}$
60.576.17-60.hm.1.3 $60$ $2$ $2$ $17$ $4$ $1^{4}\cdot2^{3}$
60.576.17-60.js.2.5 $60$ $2$ $2$ $17$ $3$ $1^{6}\cdot2^{2}$
60.576.17-60.ju.2.7 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
60.576.17-60.kf.2.5 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
60.576.17-60.kg.2.7 $60$ $2$ $2$ $17$ $4$ $1^{6}\cdot2^{2}$
60.576.17-60.ki.1.6 $60$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
60.576.17-60.kk.1.8 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
60.576.17-60.kv.1.12 $60$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
60.576.17-60.kw.1.16 $60$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
60.864.25-60.jl.2.14 $60$ $3$ $3$ $25$ $0$ $1^{8}\cdot2^{5}$
60.1440.43-60.pn.1.15 $60$ $5$ $5$ $43$ $0$ $1^{16}\cdot2^{10}$