Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot4\cdot5^{2}\cdot20$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.349 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&15\\4&37\end{bmatrix}$, $\begin{bmatrix}7&15\\12&53\end{bmatrix}$, $\begin{bmatrix}9&40\\20&7\end{bmatrix}$, $\begin{bmatrix}37&5\\32&33\end{bmatrix}$, $\begin{bmatrix}41&50\\36&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.36.1.c.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $4$ |
| Cyclic 60-torsion field degree: | $64$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + 4x + 4 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:-2:1)$, $(-1:0:1)$, $(0:2:1)$, $(4:-10:1)$, $(0:1:0)$, $(4:10:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{2}y^{17}-1312x^{2}y^{15}z^{2}+151776x^{2}y^{14}z^{3}-3975880x^{2}y^{13}z^{4}+28829600x^{2}y^{12}z^{5}-247751440x^{2}y^{11}z^{6}-173455080x^{2}y^{10}z^{7}+28774520006x^{2}y^{9}z^{8}-12179629920x^{2}y^{8}z^{9}-149788365792x^{2}y^{7}z^{10}-2772139991360x^{2}y^{6}z^{11}+4965292102964x^{2}y^{5}z^{12}+31406365793072x^{2}y^{4}z^{13}-80531644416880x^{2}y^{3}z^{14}+4823382509736x^{2}y^{2}z^{15}+103593101229955x^{2}yz^{16}-57880589893632x^{2}z^{17}-26xy^{17}z+744xy^{16}z^{2}-54xy^{15}z^{3}-386880xy^{14}z^{4}+24484860xy^{13}z^{5}-343178480xy^{12}z^{6}+1737396299xy^{11}z^{7}-25473116640xy^{10}z^{8}+115282216130xy^{9}z^{9}-78595531080xy^{8}z^{10}+2651296456137xy^{7}z^{11}-14533740581632xy^{6}z^{12}+5648682909518xy^{5}z^{13}+40724091375240xy^{4}z^{14}+82422855106535xy^{3}z^{15}-395278188281856xy^{2}z^{16}+379841371176960xyz^{17}-91644267331584xz^{18}+272y^{17}z^{2}-17112y^{16}z^{3}+208510y^{15}z^{4}-441936y^{14}z^{5}-47936425y^{13}z^{6}+1709716400y^{12}z^{7}-9722411027y^{11}z^{8}+36842629584y^{10}z^{9}-424431181416y^{9}z^{10}+1246167350040y^{8}z^{11}+2144162763411y^{7}z^{12}-707096456984y^{6}z^{13}-54920889346691y^{5}z^{14}+92232176328248y^{4}z^{15}+128423460860505y^{3}z^{16}-404924953189728y^{2}z^{17}+276248269946380yz^{18}-33763677437952z^{19}}{z^{2}(204x^{2}y^{14}z+9860x^{2}y^{12}z^{3}+113535x^{2}y^{10}z^{5}-2520540x^{2}y^{8}z^{7}-12331080x^{2}y^{6}z^{9}+92274590x^{2}y^{4}z^{11}+16777241x^{2}y^{2}z^{13}-201326592x^{2}z^{15}+xy^{16}-520xy^{14}z^{2}+970xy^{12}z^{4}+689300xy^{10}z^{6}+2704835xy^{8}z^{8}-58510560xy^{6}z^{10}+10485765xy^{4}z^{12}+486539264xy^{2}z^{14}-318767104xz^{16}-23y^{16}z-594y^{14}z^{3}-70930y^{12}z^{5}-312958y^{10}z^{7}+10685375y^{8}z^{9}-18076735y^{6}z^{11}-165675405y^{4}z^{13}+452984932y^{2}z^{15}-117440512z^{17})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.12.0-4.c.1.1 | $60$ | $6$ | $6$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.1-20.f.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-20.f.2.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-20.g.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-20.g.2.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.m.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.m.2.8 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.n.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.1-60.n.2.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.144.3-20.b.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-20.l.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-20.o.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-20.p.1.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-20.s.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-20.s.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-20.t.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-20.t.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-60.es.1.5 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.et.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fe.1.7 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.ff.1.3 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.hu.1.6 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-60.hu.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-60.hv.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.144.3-60.hv.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.216.7-60.c.1.27 | $60$ | $3$ | $3$ | $7$ | $0$ | $1^{6}$ |
| 60.288.7-60.gx.1.18 | $60$ | $4$ | $4$ | $7$ | $0$ | $1^{6}$ |
| 60.360.7-20.i.1.6 | $60$ | $5$ | $5$ | $7$ | $0$ | $1^{6}$ |
| 120.144.1-40.s.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-40.s.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-40.v.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-40.v.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bq.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bq.2.22 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bt.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bt.2.22 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-40.h.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bi.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bq.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bt.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bw.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bw.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bx.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bx.1.44 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.by.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.by.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bz.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.bz.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ca.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ca.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ca.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ca.2.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cb.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cb.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cb.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cb.2.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cc.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cc.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cc.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cc.2.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cd.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cd.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cd.2.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cd.2.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ce.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ce.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cf.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cf.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cg.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cg.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ch.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.ch.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.co.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.co.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cr.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-40.cr.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bec.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bef.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bgs.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bgv.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bye.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bye.1.37 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byf.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byf.1.45 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byg.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byg.1.33 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byh.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byh.1.41 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byi.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byi.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byi.2.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byi.2.35 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byj.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byj.1.47 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byj.2.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byj.2.43 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byk.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byk.1.34 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byk.2.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byk.2.38 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byl.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byl.1.50 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byl.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byl.2.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bym.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bym.1.41 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byn.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byn.1.33 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byo.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byo.1.45 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byp.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byp.1.37 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgk.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgk.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgn.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgn.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 300.360.7-100.c.1.8 | $300$ | $5$ | $5$ | $7$ | $?$ | not computed |