The curve $X_{\mathrm{arith}}(5)$ over $\mathbb{C}$ is the classical modular curve $X(5)$, studied by Klein in connection with the quintic equation. Its automorphism group over $\mathbb{C}$ is $\operatorname{PSL}_2(\mathbb{F}_5) \simeq A_5$.
Invariants
| Level: | $5$ | $\SL_2$-level: | $5$ | ||||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $0 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $5^{12}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| 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: | 5H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 5.120.0.1 |
| Sutherland (S) label: | 5Cs.1.1 |
Level structure
| $\GL_2(\Z/5\Z)$-generators: | $\begin{bmatrix}1&0\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/5\Z)$-subgroup: | $C_4$ |
| Contains $-I$: | no $\quad$ (see 5.60.0.a.1 for the level structure with $-I$) |
| Cyclic 5-isogeny field degree: | $1$ |
| Cyclic 5-torsion field degree: | $1$ |
| Full 5-torsion field degree: | $4$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 60 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{60}(x^{4}+3x^{3}y-x^{2}y^{2}-3xy^{3}+y^{4})^{3}(x^{8}-4x^{7}y+7x^{6}y^{2}-2x^{5}y^{3}+15x^{4}y^{4}+2x^{3}y^{5}+7x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+x^{7}y+7x^{6}y^{2}-7x^{5}y^{3}+7x^{3}y^{5}+7x^{2}y^{6}-xy^{7}+y^{8})^{3}}{y^{5}x^{65}(x^{2}-xy-y^{2})^{5}(x^{4}-2x^{3}y+4x^{2}y^{2}-3xy^{3}+y^{4})^{5}(x^{4}+3x^{3}y+4x^{2}y^{2}+2xy^{3}+y^{4})^{5}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $5$ | $5$ | $0$ | $0$ |
| 5.24.0-5.a.2.2 | $5$ | $5$ | $5$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.