Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.0.47 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}7&6\\6&11\end{bmatrix}$, $\begin{bmatrix}11&4\\0&5\end{bmatrix}$, $\begin{bmatrix}11&5\\0&11\end{bmatrix}$, $\begin{bmatrix}11&8\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^4:D_6$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| 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 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.48.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-6.b.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.72.0-6.a.1.5 | $12$ | $3$ | $3$ | $0$ |
| 12.48.0-12.d.1.10 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.g.1.11 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.h.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.i.1.1 | $12$ | $2$ | $2$ | $0$ |
| 12.48.0-12.j.1.5 | $12$ | $2$ | $2$ | $0$ |
| 12.48.1-12.i.1.4 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1-12.j.1.8 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1-12.k.1.2 | $12$ | $2$ | $2$ | $1$ |
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ |
| 36.72.0-18.a.1.12 | $36$ | $3$ | $3$ | $0$ |
| 36.72.2-18.c.1.3 | $36$ | $3$ | $3$ | $2$ |
| 36.72.2-18.d.1.1 | $36$ | $3$ | $3$ | $2$ |
| 24.48.0-24.p.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bw.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bx.1.14 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.ca.1.10 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cb.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cc.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cd.1.14 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-24.eq.1.6 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.er.1.10 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.es.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.et.1.4 | $24$ | $2$ | $2$ | $1$ |
| 60.48.0-30.a.1.8 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-30.b.1.5 | $60$ | $2$ | $2$ | $0$ |
| 60.120.4-30.b.1.2 | $60$ | $5$ | $5$ | $4$ |
| 60.144.3-30.a.1.9 | $60$ | $6$ | $6$ | $3$ |
| 60.240.7-30.h.1.14 | $60$ | $10$ | $10$ | $7$ |
| 84.48.0-42.b.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-42.c.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.192.5-42.a.1.47 | $84$ | $8$ | $8$ | $5$ |
| 84.504.16-42.a.1.22 | $84$ | $21$ | $21$ | $16$ |
| 60.48.0-60.o.1.11 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.p.1.5 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.q.1.7 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.r.1.6 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.s.1.6 | $60$ | $2$ | $2$ | $0$ |
| 60.48.0-60.t.1.8 | $60$ | $2$ | $2$ | $0$ |
| 60.48.1-60.v.1.9 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.w.1.11 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.x.1.10 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.y.1.12 | $60$ | $2$ | $2$ | $1$ |
| 132.48.0-66.a.1.3 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-66.b.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.288.9-66.a.1.16 | $132$ | $12$ | $12$ | $9$ |
| 156.48.0-78.b.1.11 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-78.c.1.12 | $156$ | $2$ | $2$ | $0$ |
| 156.336.11-78.a.1.45 | $156$ | $14$ | $14$ | $11$ |
| 84.48.0-84.m.1.6 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.n.1.13 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.o.1.13 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.p.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.q.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.r.1.7 | $84$ | $2$ | $2$ | $0$ |
| 84.48.1-84.m.1.9 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.n.1.13 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.o.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.p.1.7 | $84$ | $2$ | $2$ | $1$ |
| 204.48.0-102.a.1.11 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-102.b.1.11 | $204$ | $2$ | $2$ | $0$ |
| 204.432.15-102.a.1.2 | $204$ | $18$ | $18$ | $15$ |
| 228.48.0-114.b.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-114.c.1.1 | $228$ | $2$ | $2$ | $0$ |
| 228.480.17-114.a.1.34 | $228$ | $20$ | $20$ | $17$ |
| 120.48.0-120.fm.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fn.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fo.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fp.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fq.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fr.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fs.1.22 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ft.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-120.iu.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iv.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iw.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.ix.1.10 | $120$ | $2$ | $2$ | $1$ |
| 132.48.0-132.m.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.n.1.9 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.o.1.11 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.p.1.2 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.q.1.5 | $132$ | $2$ | $2$ | $0$ |
| 132.48.0-132.r.1.7 | $132$ | $2$ | $2$ | $0$ |
| 132.48.1-132.m.1.10 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1-132.n.1.12 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1-132.o.1.6 | $132$ | $2$ | $2$ | $1$ |
| 132.48.1-132.p.1.8 | $132$ | $2$ | $2$ | $1$ |
| 276.48.0-138.a.1.3 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-138.b.1.4 | $276$ | $2$ | $2$ | $0$ |
| 156.48.0-156.o.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.p.1.12 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.q.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.r.1.13 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.s.1.16 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.t.1.10 | $156$ | $2$ | $2$ | $0$ |
| 156.48.1-156.m.1.8 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.n.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.o.1.4 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.p.1.6 | $156$ | $2$ | $2$ | $1$ |
| 168.48.0-168.fg.1.22 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fh.1.26 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fi.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fj.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fk.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fl.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fm.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fn.1.27 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-168.hk.1.22 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hl.1.22 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hm.1.28 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hn.1.24 | $168$ | $2$ | $2$ | $1$ |
| 204.48.0-204.o.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.p.1.3 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.q.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.r.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.s.1.11 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.t.1.12 | $204$ | $2$ | $2$ | $0$ |
| 204.48.1-204.m.1.5 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.n.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.o.1.13 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.p.1.14 | $204$ | $2$ | $2$ | $1$ |
| 228.48.0-228.m.1.4 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.n.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.o.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.p.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.q.1.5 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.r.1.11 | $228$ | $2$ | $2$ | $0$ |
| 228.48.1-228.m.1.3 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.n.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.o.1.11 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.p.1.11 | $228$ | $2$ | $2$ | $1$ |
| 264.48.0-264.fg.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fh.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fi.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fj.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fk.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fl.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fm.1.29 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fn.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-264.hk.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hl.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hm.1.26 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hn.1.26 | $264$ | $2$ | $2$ | $1$ |
| 276.48.0-276.m.1.6 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.n.1.9 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.o.1.13 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.p.1.2 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.q.1.2 | $276$ | $2$ | $2$ | $0$ |
| 276.48.0-276.r.1.6 | $276$ | $2$ | $2$ | $0$ |
| 276.48.1-276.m.1.11 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1-276.n.1.15 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1-276.o.1.4 | $276$ | $2$ | $2$ | $1$ |
| 276.48.1-276.p.1.8 | $276$ | $2$ | $2$ | $1$ |
| 312.48.0-312.fm.1.26 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fn.1.20 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fo.1.29 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fp.1.29 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fq.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fr.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fs.1.23 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ft.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-312.hk.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hl.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hm.1.23 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hn.1.27 | $312$ | $2$ | $2$ | $1$ |