Invariants
Level: | $60$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.24.0.75 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&14\\2&51\end{bmatrix}$, $\begin{bmatrix}27&26\\4&13\end{bmatrix}$, $\begin{bmatrix}29&26\\16&41\end{bmatrix}$, $\begin{bmatrix}49&8\\58&49\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 20.12.0.a.1 for the level structure with $-I$) |
Cyclic 60-isogeny field degree: | $48$ |
Cyclic 60-torsion field degree: | $768$ |
Full 60-torsion field degree: | $92160$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 480 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^2}\cdot\frac{x^{12}(25x^{4}+1280x^{2}y^{2}+65536y^{4})^{3}}{y^{4}x^{16}(5x^{2}+256y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.12.0-2.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
60.12.0-2.a.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
60.48.0-20.a.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-20.c.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-20.c.1.4 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.d.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.d.1.7 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.f.1.1 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.f.1.6 | $60$ | $2$ | $2$ | $0$ |
60.72.2-60.a.1.4 | $60$ | $3$ | $3$ | $2$ |
60.96.1-60.a.1.12 | $60$ | $4$ | $4$ | $1$ |
60.120.4-20.c.1.4 | $60$ | $5$ | $5$ | $4$ |
60.144.3-20.c.1.7 | $60$ | $6$ | $6$ | $3$ |
60.240.7-20.c.1.8 | $60$ | $10$ | $10$ | $7$ |
120.48.0-40.b.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.b.1.7 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.f.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-40.f.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.i.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.i.1.12 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.o.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.o.1.12 | $120$ | $2$ | $2$ | $0$ |