Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.198 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&54\\24&53\end{bmatrix}$, $\begin{bmatrix}23&32\\8&25\end{bmatrix}$, $\begin{bmatrix}27&8\\22&39\end{bmatrix}$, $\begin{bmatrix}39&16\\58&15\end{bmatrix}$, $\begin{bmatrix}55&16\\46&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y + 4 y^{2} + z w $ |
| $=$ | $x^{2} - 4 x y + z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + x^{2} y^{2} + 2 x^{2} y z + x^{2} z^{2} + y^{2} z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{3xz^{8}-2xz^{7}w+6xz^{5}w^{3}+2xz^{4}w^{4}+6xz^{3}w^{5}-2xzw^{7}+3xw^{8}+4yz^{8}-20yz^{7}w+16yz^{6}w^{2}-28yz^{5}w^{3}+8yz^{4}w^{4}-28yz^{3}w^{5}+16yz^{2}w^{6}-20yzw^{7}+4yw^{8}}{xz^{7}w-3xz^{6}w^{2}+8xz^{4}w^{4}-3xz^{2}w^{6}+xzw^{7}-2yz^{8}+4yz^{7}w+10yz^{6}w^{2}-40yz^{5}w^{3}+56yz^{4}w^{4}-40yz^{3}w^{5}+10yz^{2}w^{6}+4yzw^{7}-2yw^{8}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.1.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}+X^{2}Y^{2}+2X^{2}YZ+X^{2}Z^{2}+Y^{2}Z^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.36.1-6.a.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1-6.a.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.3-12.j.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.j.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.l.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.l.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.l.1.6 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.n.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.n.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.n.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.p.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.p.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.p.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.ba.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.ba.1.7 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.ba.1.8 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.bb.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bb.1.6 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bb.1.8 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bi.1.3 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bi.1.5 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bi.1.8 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.bj.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.bj.1.5 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.bj.1.7 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.360.13-60.b.1.1 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{12}$ |
| 60.432.13-60.d.1.5 | $60$ | $6$ | $6$ | $13$ | $0$ | $1^{12}$ |
| 60.720.25-60.h.1.1 | $60$ | $10$ | $10$ | $25$ | $8$ | $1^{24}$ |
| 120.144.3-24.bc.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bc.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bc.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bi.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bi.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bi.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bo.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bo.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bo.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bu.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bu.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bu.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cs.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cs.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cs.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cv.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cv.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cv.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dq.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dq.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dq.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dt.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dt.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dt.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.216.7-36.b.1.1 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |