Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.32.0.3 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}2&5\\9&1\end{bmatrix}$, $\begin{bmatrix}8&9\\9&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $A_4\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.16.0.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $6$ |
| Cyclic 12-torsion field degree: | $24$ |
| Full 12-torsion field degree: | $144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 14 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^6}\cdot\frac{x^{16}(27x^{4}+16y^{4})^{3}(243x^{4}+16y^{4})}{y^{4}x^{28}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.4.0-2.a.1.1 | $4$ | $8$ | $8$ | $0$ | $0$ |
| 3.8.0-3.a.1.1 | $3$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.16.0-6.a.1.1 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 12.16.0-6.a.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.96.2-12.a.2.8 | $12$ | $3$ | $3$ | $2$ |
| 12.96.3-12.a.1.4 | $12$ | $3$ | $3$ | $3$ |
| 12.128.1-12.a.2.4 | $12$ | $4$ | $4$ | $1$ |
| 36.96.0-36.a.2.4 | $36$ | $3$ | $3$ | $0$ |
| 36.96.2-36.a.2.2 | $36$ | $3$ | $3$ | $2$ |
| 36.96.2-36.b.2.4 | $36$ | $3$ | $3$ | $2$ |
| 36.96.3-36.a.2.3 | $36$ | $3$ | $3$ | $3$ |
| 36.96.4-36.a.2.4 | $36$ | $3$ | $3$ | $4$ |
| 60.160.4-60.c.1.3 | $60$ | $5$ | $5$ | $4$ |
| 60.192.7-60.e.2.10 | $60$ | $6$ | $6$ | $7$ |
| 60.320.11-60.i.1.16 | $60$ | $10$ | $10$ | $11$ |
| 84.96.2-84.e.2.6 | $84$ | $3$ | $3$ | $2$ |
| 84.96.2-84.f.1.5 | $84$ | $3$ | $3$ | $2$ |
| 84.256.7-84.g.2.8 | $84$ | $8$ | $8$ | $7$ |
| 132.384.15-132.g.1.16 | $132$ | $12$ | $12$ | $15$ |
| 156.96.2-156.e.1.5 | $156$ | $3$ | $3$ | $2$ |
| 156.96.2-156.f.2.6 | $156$ | $3$ | $3$ | $2$ |
| 156.448.15-156.e.2.16 | $156$ | $14$ | $14$ | $15$ |
| 228.96.2-228.e.1.3 | $228$ | $3$ | $3$ | $2$ |
| 228.96.2-228.f.1.4 | $228$ | $3$ | $3$ | $2$ |
| 252.96.2-252.g.2.7 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.h.2.6 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.i.2.7 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.j.2.6 | $252$ | $3$ | $3$ | $2$ |