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Properties

Label 60.72.1.y.2
Level $60$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $720$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.27

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}9&5\\34&9\end{bmatrix}$, $\begin{bmatrix}13&30\\22&31\end{bmatrix}$, $\begin{bmatrix}17&5\\22&11\end{bmatrix}$, $\begin{bmatrix}17&45\\14&41\end{bmatrix}$, $\begin{bmatrix}47&0\\36&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.144.1-60.y.2.1, 60.144.1-60.y.2.2, 60.144.1-60.y.2.3, 60.144.1-60.y.2.4, 60.144.1-60.y.2.5, 60.144.1-60.y.2.6, 60.144.1-60.y.2.7, 60.144.1-60.y.2.8, 60.144.1-60.y.2.9, 60.144.1-60.y.2.10, 60.144.1-60.y.2.11, 60.144.1-60.y.2.12, 60.144.1-60.y.2.13, 60.144.1-60.y.2.14, 60.144.1-60.y.2.15, 60.144.1-60.y.2.16, 120.144.1-60.y.2.1, 120.144.1-60.y.2.2, 120.144.1-60.y.2.3, 120.144.1-60.y.2.4, 120.144.1-60.y.2.5, 120.144.1-60.y.2.6, 120.144.1-60.y.2.7, 120.144.1-60.y.2.8, 120.144.1-60.y.2.9, 120.144.1-60.y.2.10, 120.144.1-60.y.2.11, 120.144.1-60.y.2.12, 120.144.1-60.y.2.13, 120.144.1-60.y.2.14, 120.144.1-60.y.2.15, 120.144.1-60.y.2.16
Cyclic 60-isogeny field degree: $8$
Cyclic 60-torsion field degree: $64$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{4}\cdot3^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 720.2.a.h

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 12x + 11 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(1:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^{10}}\cdot\frac{72x^{2}y^{22}-574776x^{2}y^{20}z^{2}+320614200x^{2}y^{18}z^{4}-54848962728x^{2}y^{16}z^{6}+4569584809680x^{2}y^{14}z^{8}-235415974047408x^{2}y^{12}z^{10}+8535592425097584x^{2}y^{10}z^{12}-225664965375472080x^{2}y^{8}z^{14}+4399386814274202216x^{2}y^{6}z^{16}-62494821628462459800x^{2}y^{4}z^{18}+589097354278266475992x^{2}y^{2}z^{20}-3329165719614239170056x^{2}z^{22}-2304xy^{22}z+5885136xy^{20}z^{3}-2172828240xy^{18}z^{5}+305650850976xy^{16}z^{7}-22957741016640xy^{14}z^{9}+1114425195634656xy^{12}z^{11}-38682776664546912xy^{10}z^{13}+987223030451452800xy^{8}z^{15}-18700052752321791552xy^{6}z^{17}+257847072974960026320xy^{4}z^{19}-2372725643097983734224xy^{2}z^{21}+12831023347362666190944xz^{23}-y^{24}+43596y^{22}z^{2}-44211258y^{20}z^{4}+10017903420y^{18}z^{6}-1002905163063y^{16}z^{8}+59226178650264y^{14}z^{10}-2416961634862284y^{12}z^{12}+72043409302233048y^{10}z^{14}-1597960428035320575y^{8}z^{16}+26483167968113120796y^{6}z^{18}-312387771127587276186y^{4}z^{20}+2469549654005870130156y^{2}z^{22}-9502007722383724020009z^{24}}{z^{6}y^{4}(y^{2}-27z^{2})^{2}(x^{2}y^{8}-3996x^{2}y^{6}z^{2}+788778x^{2}y^{4}z^{4}-41019372x^{2}y^{2}z^{6}+619128765x^{2}z^{8}-32xy^{8}z+29862xy^{6}z^{3}-4092606xy^{4}z^{5}+177107634xy^{2}z^{7}-2386170090xz^{9}+436y^{8}z^{2}-148824y^{6}z^{4}+10947393y^{4}z^{6}-263634102y^{2}z^{8}+1767041325z^{10})}$

Modular covers

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Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\pm1}(5)$ $5$ $6$ $6$ $0$ $0$ full Jacobian
12.6.0.b.1 $12$ $12$ $12$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\pm1}(10)$ $10$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.d.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.36.1.w.1 $60$ $2$ $2$ $1$ $0$ 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
60.144.5.y.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.ce.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.ia.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.ie.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.kp.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.kr.1 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.kz.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.lb.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.216.13.cv.2 $60$ $3$ $3$ $13$ $1$ $1^{6}\cdot2^{3}$
60.288.13.jx.2 $60$ $4$ $4$ $13$ $1$ $1^{6}\cdot2^{3}$
60.360.13.r.1 $60$ $5$ $5$ $13$ $2$ $1^{6}\cdot2^{3}$
120.144.5.gu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.pf.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.clo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cmn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ddc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ddq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dfw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dgk.1 $120$ $2$ $2$ $5$ $?$ not computed
300.360.13.k.2 $300$ $5$ $5$ $13$ $?$ not computed