Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.8.0.9 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 310 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{x^{8}(x^{2}-54y^{2})(x^{2}-6y^{2})^{3}}{y^{6}x^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
| 8.2.0.a.1 | $8$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
| 8.2.0.a.1 | $8$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.24.0.p.1 | $24$ | $3$ | $3$ | $0$ |
| 24.24.1.bx.1 | $24$ | $3$ | $3$ | $1$ |
| 24.32.1.a.1 | $24$ | $4$ | $4$ | $1$ |
| 72.24.0.a.1 | $72$ | $3$ | $3$ | $0$ |
| 72.24.1.a.1 | $72$ | $3$ | $3$ | $1$ |
| 72.24.2.a.1 | $72$ | $3$ | $3$ | $2$ |
| 120.40.2.a.1 | $120$ | $5$ | $5$ | $2$ |
| 120.48.3.c.1 | $120$ | $6$ | $6$ | $3$ |
| 120.80.5.a.1 | $120$ | $10$ | $10$ | $5$ |
| 168.64.3.a.1 | $168$ | $8$ | $8$ | $3$ |
| 168.168.12.a.1 | $168$ | $21$ | $21$ | $12$ |
| 168.224.15.a.1 | $168$ | $28$ | $28$ | $15$ |
| 264.96.7.a.1 | $264$ | $12$ | $12$ | $7$ |
| 312.112.7.a.1 | $312$ | $14$ | $14$ | $7$ |