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 | $1^{2}\cdot3^{2}\cdot4\cdot12$ | 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: | 12E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.0.59 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&7\\6&5\end{bmatrix}$, $\begin{bmatrix}5&0\\6&7\end{bmatrix}$, $\begin{bmatrix}7&7\\0&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $D_6.D_4$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.f.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}+4y^{2})^{3}(3x^{6}+12x^{4}y^{2}+144x^{2}y^{4}+64y^{6})^{3}}{y^{4}x^{36}(x^{2}+4y^{2})^{3}(9x^{2}+4y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.96.1-12.a.1.12 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.e.1.2 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.i.1.2 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.j.1.1 | $12$ | $2$ | $2$ | $1$ |
| 12.144.1-12.d.1.3 | $12$ | $3$ | $3$ | $1$ |
| 24.96.0-24.bq.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bq.2.14 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.br.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.br.2.10 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.cf.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.dq.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.ie.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.ih.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.2-24.d.1.6 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.d.2.3 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.e.1.5 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.e.2.1 | $24$ | $2$ | $2$ | $2$ |
| 36.144.1-36.b.1.4 | $36$ | $3$ | $3$ | $1$ |
| 36.144.4-36.c.1.5 | $36$ | $3$ | $3$ | $4$ |
| 36.144.4-36.e.1.7 | $36$ | $3$ | $3$ | $4$ |
| 60.96.1-60.i.1.6 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.j.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.m.1.3 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.n.1.1 | $60$ | $2$ | $2$ | $1$ |
| 60.240.8-60.n.1.1 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.gw.1.17 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.bp.1.7 | $60$ | $10$ | $10$ | $15$ |
| 84.96.1-84.i.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.j.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.m.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.n.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.384.11-84.bl.1.1 | $84$ | $8$ | $8$ | $11$ |
| 120.96.0-120.do.1.25 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.do.2.27 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dp.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dp.2.21 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.yw.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.yz.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zi.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.zl.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.f.1.8 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.f.2.6 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.g.1.4 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.g.2.3 | $120$ | $2$ | $2$ | $2$ |
| 132.96.1-132.i.1.2 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.j.1.1 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.m.1.2 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.n.1.1 | $132$ | $2$ | $2$ | $1$ |
| 156.96.1-156.i.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.j.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.m.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.n.1.1 | $156$ | $2$ | $2$ | $1$ |
| 168.96.0-168.dm.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dm.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dn.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dn.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.yu.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.yx.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zj.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-168.d.1.29 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.d.2.25 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.e.1.29 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.e.2.25 | $168$ | $2$ | $2$ | $2$ |
| 204.96.1-204.i.1.4 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.j.1.1 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.m.1.2 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.n.1.1 | $204$ | $2$ | $2$ | $1$ |
| 228.96.1-228.i.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.j.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.m.1.1 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.n.1.1 | $228$ | $2$ | $2$ | $1$ |
| 264.96.0-264.dm.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dm.2.23 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dn.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dn.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.yu.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.yx.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zg.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.zj.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-264.d.1.18 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.d.2.18 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.e.1.17 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.e.2.9 | $264$ | $2$ | $2$ | $2$ |
| 276.96.1-276.i.1.2 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.j.1.1 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.m.1.2 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.n.1.1 | $276$ | $2$ | $2$ | $1$ |
| 312.96.0-312.do.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.do.2.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dp.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dp.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.yw.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.yz.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zi.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zl.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-312.f.1.25 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.f.2.29 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.g.1.29 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.g.2.25 | $312$ | $2$ | $2$ | $2$ |