Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 6I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.0.80 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&5\\0&11\end{bmatrix}$, $\begin{bmatrix}1&11\\0&7\end{bmatrix}$, $\begin{bmatrix}5&4\\6&5\end{bmatrix}$, $\begin{bmatrix}5&10\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $D_4\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.d.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $96$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 84 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(3x+y)^{24}(12x^{2}-y^{2})^{3}(171072x^{6}+311040x^{5}y+226800x^{4}y^{2}+86400x^{3}y^{3}+18540x^{2}y^{4}+2160xy^{5}+109y^{6})^{3}}{(2x+y)^{2}(3x+y)^{24}(6x+y)^{6}(20x^{2}+8xy+y^{2})^{2}(36x^{2}+24xy+5y^{2})^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-6.a.1.12 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.96.0-12.b.1.6 | $12$ | $2$ | $2$ | $0$ |
| 12.96.0-12.b.2.4 | $12$ | $2$ | $2$ | $0$ |
| 12.96.1-12.e.1.5 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.f.1.2 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.g.1.6 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.h.1.7 | $12$ | $2$ | $2$ | $1$ |
| 12.96.2-12.b.1.1 | $12$ | $2$ | $2$ | $2$ |
| 12.96.2-12.b.2.3 | $12$ | $2$ | $2$ | $2$ |
| 12.144.1-12.b.1.8 | $12$ | $3$ | $3$ | $1$ |
| 24.96.0-24.bh.1.9 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bh.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.dp.1.11 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.dv.1.11 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.eo.1.11 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.er.1.11 | $24$ | $2$ | $2$ | $1$ |
| 24.96.2-24.c.1.1 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.c.2.5 | $24$ | $2$ | $2$ | $2$ |
| 36.144.1-36.a.1.4 | $36$ | $3$ | $3$ | $1$ |
| 36.144.4-36.a.1.4 | $36$ | $3$ | $3$ | $4$ |
| 36.144.4-36.b.1.3 | $36$ | $3$ | $3$ | $4$ |
| 60.96.0-60.b.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.b.2.4 | $60$ | $2$ | $2$ | $0$ |
| 60.96.1-60.e.1.2 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.f.1.10 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.g.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.h.1.11 | $60$ | $2$ | $2$ | $1$ |
| 60.96.2-60.b.1.14 | $60$ | $2$ | $2$ | $2$ |
| 60.96.2-60.b.2.14 | $60$ | $2$ | $2$ | $2$ |
| 60.240.8-60.k.1.23 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.fm.1.32 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.bk.1.48 | $60$ | $10$ | $10$ | $15$ |
| 84.96.0-84.b.1.10 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.b.2.10 | $84$ | $2$ | $2$ | $0$ |
| 84.96.1-84.e.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.f.1.10 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.g.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.h.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.96.2-84.b.1.9 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.b.2.9 | $84$ | $2$ | $2$ | $2$ |
| 84.384.11-84.z.1.47 | $84$ | $8$ | $8$ | $11$ |
| 120.96.0-120.df.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.df.2.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.jw.1.18 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.jz.1.22 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.kc.1.19 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.kf.1.21 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.e.1.17 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.e.2.9 | $120$ | $2$ | $2$ | $2$ |
| 132.96.0-132.b.1.3 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.b.2.9 | $132$ | $2$ | $2$ | $0$ |
| 132.96.1-132.e.1.5 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.f.1.10 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.g.1.7 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.h.1.11 | $132$ | $2$ | $2$ | $1$ |
| 132.96.2-132.b.1.3 | $132$ | $2$ | $2$ | $2$ |
| 132.96.2-132.b.2.5 | $132$ | $2$ | $2$ | $2$ |
| 156.96.0-156.b.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.b.2.11 | $156$ | $2$ | $2$ | $0$ |
| 156.96.1-156.e.1.6 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.f.1.12 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.g.1.7 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.h.1.9 | $156$ | $2$ | $2$ | $1$ |
| 156.96.2-156.b.1.16 | $156$ | $2$ | $2$ | $2$ |
| 156.96.2-156.b.2.15 | $156$ | $2$ | $2$ | $2$ |
| 168.96.0-168.dd.1.22 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dd.2.26 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.jw.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.jz.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kc.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kf.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-168.c.1.11 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.c.2.13 | $168$ | $2$ | $2$ | $2$ |
| 204.96.0-204.b.1.7 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.b.2.11 | $204$ | $2$ | $2$ | $0$ |
| 204.96.1-204.e.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.f.1.11 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.g.1.3 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.h.1.10 | $204$ | $2$ | $2$ | $1$ |
| 204.96.2-204.b.1.15 | $204$ | $2$ | $2$ | $2$ |
| 204.96.2-204.b.2.11 | $204$ | $2$ | $2$ | $2$ |
| 228.96.0-228.b.1.10 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.b.2.10 | $228$ | $2$ | $2$ | $0$ |
| 228.96.1-228.e.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.f.1.10 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.g.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.h.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.2-228.b.1.5 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.b.2.9 | $228$ | $2$ | $2$ | $2$ |
| 264.96.0-264.dd.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dd.2.18 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.jw.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.jz.1.23 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.kc.1.11 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.kf.1.22 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-264.c.1.5 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.c.2.9 | $264$ | $2$ | $2$ | $2$ |
| 276.96.0-276.b.1.9 | $276$ | $2$ | $2$ | $0$ |
| 276.96.0-276.b.2.3 | $276$ | $2$ | $2$ | $0$ |
| 276.96.1-276.e.1.9 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.f.1.10 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.g.1.10 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.h.1.11 | $276$ | $2$ | $2$ | $1$ |
| 276.96.2-276.b.1.5 | $276$ | $2$ | $2$ | $2$ |
| 276.96.2-276.b.2.5 | $276$ | $2$ | $2$ | $2$ |
| 312.96.0-312.df.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.df.2.21 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.jw.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.jz.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.kc.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.kf.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-312.e.1.19 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.e.2.21 | $312$ | $2$ | $2$ | $2$ |