Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $4^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 4D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.8.0.7 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&0\\7&7\end{bmatrix}$, $\begin{bmatrix}5&1\\1&4\end{bmatrix}$, $\begin{bmatrix}8&9\\11&4\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_6.\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 139 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^{10}\cdot3^6\,\frac{y^{3}(12x-y)^{3}(18x-y)^{8}(36x+y)(72x-7y)}{(18x-y)^{8}(1296x^{2}-144xy+y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 6.2.0.a.1 | $6$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.24.0.k.1 | $12$ | $3$ | $3$ | $0$ |
| 12.24.1.h.1 | $12$ | $3$ | $3$ | $1$ |
| 12.24.2.c.1 | $12$ | $3$ | $3$ | $2$ |
| 12.32.1.c.1 | $12$ | $4$ | $4$ | $1$ |
| 24.32.1.e.1 | $24$ | $4$ | $4$ | $1$ |
| 36.216.14.c.1 | $36$ | $27$ | $27$ | $14$ |
| 60.40.2.c.1 | $60$ | $5$ | $5$ | $2$ |
| 60.48.3.u.1 | $60$ | $6$ | $6$ | $3$ |
| 60.80.5.c.1 | $60$ | $10$ | $10$ | $5$ |
| 84.64.3.c.1 | $84$ | $8$ | $8$ | $3$ |
| 84.168.12.c.1 | $84$ | $21$ | $21$ | $12$ |
| 84.224.15.c.1 | $84$ | $28$ | $28$ | $15$ |
| 132.96.7.c.1 | $132$ | $12$ | $12$ | $7$ |
| 156.112.7.c.1 | $156$ | $14$ | $14$ | $7$ |
| 204.144.11.c.1 | $204$ | $18$ | $18$ | $11$ |
| 228.160.11.c.1 | $228$ | $20$ | $20$ | $11$ |
| 276.192.15.c.1 | $276$ | $24$ | $24$ | $15$ |