Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.0.32 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&11\\8&3\end{bmatrix}$, $\begin{bmatrix}5&5\\4&3\end{bmatrix}$, $\begin{bmatrix}7&2\\4&3\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.d.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $192$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{(4x-y)^{12}(64x^{2}-32xy+y^{2})^{3}(64x^{2}+32xy+y^{2})^{3}}{y^{2}x^{2}(4x-y)^{12}(64x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-4.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.12.0-4.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.48.0-4.c.1.1 | $12$ | $2$ | $2$ | $0$ |
| 24.48.0-8.p.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.q.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.t.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.w.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.x.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-8.m.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-8.n.1.2 | $24$ | $2$ | $2$ | $1$ |
| 12.48.0-12.e.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.72.2-12.p.1.8 | $12$ | $3$ | $3$ | $2$ |
| 12.96.1-12.h.1.5 | $12$ | $4$ | $4$ | $1$ |
| 60.48.0-20.d.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.120.4-20.h.1.4 | $60$ | $5$ | $5$ | $4$ |
| 60.144.3-20.l.1.8 | $60$ | $6$ | $6$ | $3$ |
| 60.240.7-20.p.1.6 | $60$ | $10$ | $10$ | $7$ |
| 24.48.0-24.u.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.v.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.z.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bc.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bd.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-24.m.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.n.1.4 | $24$ | $2$ | $2$ | $1$ |
| 84.48.0-28.d.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.192.5-28.h.1.9 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-28.p.1.7 | $84$ | $21$ | $21$ | $16$ |
| 36.648.22-36.t.1.4 | $36$ | $27$ | $27$ | $22$ |
| 120.48.0-40.v.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.w.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.z.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.be.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bf.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-40.m.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.n.1.8 | $120$ | $2$ | $2$ | $1$ |
| 132.48.0-44.d.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.288.9-44.h.1.8 | $132$ | $12$ | $12$ | $9$ |
| 156.48.0-52.d.1.2 | $156$ | $2$ | $2$ | $0$ |
| 156.336.11-52.l.1.5 | $156$ | $14$ | $14$ | $11$ |
| 168.48.0-56.t.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.u.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.x.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.ba.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bb.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-56.m.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-56.n.1.4 | $168$ | $2$ | $2$ | $1$ |
| 60.48.0-60.h.1.3 | $60$ | $2$ | $2$ | $0$ |
| 204.48.0-68.d.1.1 | $204$ | $2$ | $2$ | $0$ |
| 204.432.15-68.l.1.5 | $204$ | $18$ | $18$ | $15$ |
| 228.48.0-76.d.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.480.17-76.h.1.10 | $228$ | $20$ | $20$ | $17$ |
| 84.48.0-84.h.1.1 | $84$ | $2$ | $2$ | $0$ |
| 264.48.0-88.t.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.u.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.x.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.ba.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bb.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-88.m.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-88.n.1.4 | $264$ | $2$ | $2$ | $1$ |
| 276.48.0-92.d.1.1 | $276$ | $2$ | $2$ | $0$ |
| 312.48.0-104.v.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.w.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.z.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.be.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bf.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-104.m.1.4 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-104.n.1.8 | $312$ | $2$ | $2$ | $1$ |
| 120.48.0-120.bh.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bi.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bl.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bq.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.br.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-120.m.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.n.1.6 | $120$ | $2$ | $2$ | $1$ |
| 132.48.0-132.h.1.1 | $132$ | $2$ | $2$ | $0$ |
| 156.48.0-156.h.1.1 | $156$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bf.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bg.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bj.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bm.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bn.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-168.m.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.n.1.4 | $168$ | $2$ | $2$ | $1$ |
| 204.48.0-204.h.1.4 | $204$ | $2$ | $2$ | $0$ |
| 228.48.0-228.h.1.1 | $228$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bf.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bg.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bj.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bm.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bn.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-264.m.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.n.1.12 | $264$ | $2$ | $2$ | $1$ |
| 276.48.0-276.h.1.1 | $276$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bh.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bi.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bl.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bq.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.br.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-312.m.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.n.1.4 | $312$ | $2$ | $2$ | $1$ |