Invariants
Level: | $6$ | $\SL_2$-level: | $6$ | ||||
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: | 6.24.0.8 |
Level structure
$\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}1&5\\0&5\end{bmatrix}$, $\begin{bmatrix}5&2\\0&1\end{bmatrix}$ |
$\GL_2(\Z/6\Z)$-subgroup: | $D_6$ |
Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
Cyclic 6-isogeny field degree: | $1$ |
Cyclic 6-torsion field degree: | $2$ |
Full 6-torsion field degree: | $12$ |
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 minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
6.8.0-3.a.1.2 | $6$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ |
6.48.0-6.b.1.2 | $6$ | $2$ | $2$ | $0$ |
6.72.0-6.a.1.2 | $6$ | $3$ | $3$ | $0$ |
12.48.0-12.d.1.9 | $12$ | $2$ | $2$ | $0$ |
12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ |
12.48.0-12.g.1.6 | $12$ | $2$ | $2$ | $0$ |
12.48.0-12.h.1.3 | $12$ | $2$ | $2$ | $0$ |
12.48.0-12.i.1.3 | $12$ | $2$ | $2$ | $0$ |
12.48.0-12.j.1.3 | $12$ | $2$ | $2$ | $0$ |
12.48.1-12.i.1.3 | $12$ | $2$ | $2$ | $1$ |
12.48.1-12.j.1.3 | $12$ | $2$ | $2$ | $1$ |
12.48.1-12.k.1.3 | $12$ | $2$ | $2$ | $1$ |
12.48.1-12.l.1.5 | $12$ | $2$ | $2$ | $1$ |
18.72.0-18.a.1.3 | $18$ | $3$ | $3$ | $0$ |
18.72.2-18.c.1.2 | $18$ | $3$ | $3$ | $2$ |
18.72.2-18.d.1.3 | $18$ | $3$ | $3$ | $2$ |
24.48.0-24.p.1.3 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.y.1.3 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.bw.1.6 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.bx.1.6 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.ca.1.9 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.cb.1.5 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.cc.1.4 | $24$ | $2$ | $2$ | $0$ |
24.48.0-24.cd.1.6 | $24$ | $2$ | $2$ | $0$ |
24.48.1-24.eq.1.5 | $24$ | $2$ | $2$ | $1$ |
24.48.1-24.er.1.9 | $24$ | $2$ | $2$ | $1$ |
24.48.1-24.es.1.3 | $24$ | $2$ | $2$ | $1$ |
24.48.1-24.et.1.3 | $24$ | $2$ | $2$ | $1$ |
30.48.0-30.a.1.3 | $30$ | $2$ | $2$ | $0$ |
30.48.0-30.b.1.1 | $30$ | $2$ | $2$ | $0$ |
30.120.4-30.b.1.1 | $30$ | $5$ | $5$ | $4$ |
30.144.3-30.a.1.3 | $30$ | $6$ | $6$ | $3$ |
30.240.7-30.h.1.7 | $30$ | $10$ | $10$ | $7$ |
42.48.0-42.b.1.2 | $42$ | $2$ | $2$ | $0$ |
42.48.0-42.c.1.1 | $42$ | $2$ | $2$ | $0$ |
42.192.5-42.a.1.1 | $42$ | $8$ | $8$ | $5$ |
42.504.16-42.a.1.1 | $42$ | $21$ | $21$ | $16$ |
42.672.21-42.a.1.7 | $42$ | $28$ | $28$ | $21$ |
60.48.0-60.o.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.p.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.q.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.r.1.9 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.s.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.0-60.t.1.2 | $60$ | $2$ | $2$ | $0$ |
60.48.1-60.v.1.2 | $60$ | $2$ | $2$ | $1$ |
60.48.1-60.w.1.2 | $60$ | $2$ | $2$ | $1$ |
60.48.1-60.x.1.2 | $60$ | $2$ | $2$ | $1$ |
60.48.1-60.y.1.2 | $60$ | $2$ | $2$ | $1$ |
66.48.0-66.a.1.3 | $66$ | $2$ | $2$ | $0$ |
66.48.0-66.b.1.1 | $66$ | $2$ | $2$ | $0$ |
66.288.9-66.a.1.5 | $66$ | $12$ | $12$ | $9$ |
66.1320.46-66.a.1.3 | $66$ | $55$ | $55$ | $46$ |
66.1320.46-66.b.1.5 | $66$ | $55$ | $55$ | $46$ |
66.1584.55-66.a.1.3 | $66$ | $66$ | $66$ | $55$ |
78.48.0-78.b.1.2 | $78$ | $2$ | $2$ | $0$ |
78.48.0-78.c.1.1 | $78$ | $2$ | $2$ | $0$ |
78.336.11-78.a.1.8 | $78$ | $14$ | $14$ | $11$ |
84.48.0-84.m.1.7 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.n.1.15 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.o.1.15 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.p.1.13 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.q.1.11 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.r.1.15 | $84$ | $2$ | $2$ | $0$ |
84.48.1-84.m.1.12 | $84$ | $2$ | $2$ | $1$ |
84.48.1-84.n.1.16 | $84$ | $2$ | $2$ | $1$ |
84.48.1-84.o.1.16 | $84$ | $2$ | $2$ | $1$ |
84.48.1-84.p.1.16 | $84$ | $2$ | $2$ | $1$ |
102.48.0-102.a.1.3 | $102$ | $2$ | $2$ | $0$ |
102.48.0-102.b.1.1 | $102$ | $2$ | $2$ | $0$ |
102.432.15-102.a.1.3 | $102$ | $18$ | $18$ | $15$ |
114.48.0-114.b.1.2 | $114$ | $2$ | $2$ | $0$ |
114.48.0-114.c.1.1 | $114$ | $2$ | $2$ | $0$ |
114.480.17-114.a.1.5 | $114$ | $20$ | $20$ | $17$ |
120.48.0-120.fm.1.11 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fn.1.7 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fo.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fp.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fq.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fr.1.3 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.fs.1.18 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.ft.1.4 | $120$ | $2$ | $2$ | $0$ |
120.48.1-120.iu.1.7 | $120$ | $2$ | $2$ | $1$ |
120.48.1-120.iv.1.11 | $120$ | $2$ | $2$ | $1$ |
120.48.1-120.iw.1.3 | $120$ | $2$ | $2$ | $1$ |
120.48.1-120.ix.1.9 | $120$ | $2$ | $2$ | $1$ |
132.48.0-132.m.1.11 | $132$ | $2$ | $2$ | $0$ |
132.48.0-132.n.1.13 | $132$ | $2$ | $2$ | $0$ |
132.48.0-132.o.1.13 | $132$ | $2$ | $2$ | $0$ |
132.48.0-132.p.1.11 | $132$ | $2$ | $2$ | $0$ |
132.48.0-132.q.1.13 | $132$ | $2$ | $2$ | $0$ |
132.48.0-132.r.1.13 | $132$ | $2$ | $2$ | $0$ |
132.48.1-132.m.1.13 | $132$ | $2$ | $2$ | $1$ |
132.48.1-132.n.1.13 | $132$ | $2$ | $2$ | $1$ |
132.48.1-132.o.1.13 | $132$ | $2$ | $2$ | $1$ |
132.48.1-132.p.1.13 | $132$ | $2$ | $2$ | $1$ |
138.48.0-138.a.1.2 | $138$ | $2$ | $2$ | $0$ |
138.48.0-138.b.1.1 | $138$ | $2$ | $2$ | $0$ |
156.48.0-156.o.1.7 | $156$ | $2$ | $2$ | $0$ |
156.48.0-156.p.1.6 | $156$ | $2$ | $2$ | $0$ |
156.48.0-156.q.1.6 | $156$ | $2$ | $2$ | $0$ |
156.48.0-156.r.1.8 | $156$ | $2$ | $2$ | $0$ |
156.48.0-156.s.1.6 | $156$ | $2$ | $2$ | $0$ |
156.48.0-156.t.1.6 | $156$ | $2$ | $2$ | $0$ |
156.48.1-156.m.1.5 | $156$ | $2$ | $2$ | $1$ |
156.48.1-156.n.1.5 | $156$ | $2$ | $2$ | $1$ |
156.48.1-156.o.1.5 | $156$ | $2$ | $2$ | $1$ |
156.48.1-156.p.1.5 | $156$ | $2$ | $2$ | $1$ |
168.48.0-168.fg.1.23 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fh.1.27 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fi.1.31 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fj.1.31 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fk.1.27 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fl.1.27 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fm.1.31 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.fn.1.31 | $168$ | $2$ | $2$ | $0$ |
168.48.1-168.hk.1.23 | $168$ | $2$ | $2$ | $1$ |
168.48.1-168.hl.1.23 | $168$ | $2$ | $2$ | $1$ |
168.48.1-168.hm.1.27 | $168$ | $2$ | $2$ | $1$ |
168.48.1-168.hn.1.23 | $168$ | $2$ | $2$ | $1$ |
174.48.0-174.a.1.3 | $174$ | $2$ | $2$ | $0$ |
174.48.0-174.b.1.1 | $174$ | $2$ | $2$ | $0$ |
186.48.0-186.b.1.2 | $186$ | $2$ | $2$ | $0$ |
186.48.0-186.c.1.1 | $186$ | $2$ | $2$ | $0$ |
204.48.0-204.o.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.0-204.p.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.0-204.q.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.0-204.r.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.0-204.s.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.0-204.t.1.9 | $204$ | $2$ | $2$ | $0$ |
204.48.1-204.m.1.9 | $204$ | $2$ | $2$ | $1$ |
204.48.1-204.n.1.9 | $204$ | $2$ | $2$ | $1$ |
204.48.1-204.o.1.9 | $204$ | $2$ | $2$ | $1$ |
204.48.1-204.p.1.9 | $204$ | $2$ | $2$ | $1$ |
210.48.0-210.a.1.7 | $210$ | $2$ | $2$ | $0$ |
210.48.0-210.b.1.5 | $210$ | $2$ | $2$ | $0$ |
222.48.0-222.b.1.2 | $222$ | $2$ | $2$ | $0$ |
222.48.0-222.c.1.1 | $222$ | $2$ | $2$ | $0$ |
228.48.0-228.m.1.7 | $228$ | $2$ | $2$ | $0$ |
228.48.0-228.n.1.15 | $228$ | $2$ | $2$ | $0$ |
228.48.0-228.o.1.15 | $228$ | $2$ | $2$ | $0$ |
228.48.0-228.p.1.7 | $228$ | $2$ | $2$ | $0$ |
228.48.0-228.q.1.7 | $228$ | $2$ | $2$ | $0$ |
228.48.0-228.r.1.15 | $228$ | $2$ | $2$ | $0$ |
228.48.1-228.m.1.8 | $228$ | $2$ | $2$ | $1$ |
228.48.1-228.n.1.16 | $228$ | $2$ | $2$ | $1$ |
228.48.1-228.o.1.16 | $228$ | $2$ | $2$ | $1$ |
228.48.1-228.p.1.16 | $228$ | $2$ | $2$ | $1$ |
246.48.0-246.a.1.2 | $246$ | $2$ | $2$ | $0$ |
246.48.0-246.b.1.1 | $246$ | $2$ | $2$ | $0$ |
258.48.0-258.b.1.2 | $258$ | $2$ | $2$ | $0$ |
258.48.0-258.c.1.1 | $258$ | $2$ | $2$ | $0$ |
264.48.0-264.fg.1.25 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fh.1.25 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fi.1.21 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fj.1.21 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fk.1.7 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fl.1.11 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fm.1.25 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.fn.1.21 | $264$ | $2$ | $2$ | $0$ |
264.48.1-264.hk.1.21 | $264$ | $2$ | $2$ | $1$ |
264.48.1-264.hl.1.19 | $264$ | $2$ | $2$ | $1$ |
264.48.1-264.hm.1.25 | $264$ | $2$ | $2$ | $1$ |
264.48.1-264.hn.1.25 | $264$ | $2$ | $2$ | $1$ |
276.48.0-276.m.1.7 | $276$ | $2$ | $2$ | $0$ |
276.48.0-276.n.1.10 | $276$ | $2$ | $2$ | $0$ |
276.48.0-276.o.1.10 | $276$ | $2$ | $2$ | $0$ |
276.48.0-276.p.1.11 | $276$ | $2$ | $2$ | $0$ |
276.48.0-276.q.1.10 | $276$ | $2$ | $2$ | $0$ |
276.48.0-276.r.1.10 | $276$ | $2$ | $2$ | $0$ |
276.48.1-276.m.1.10 | $276$ | $2$ | $2$ | $1$ |
276.48.1-276.n.1.10 | $276$ | $2$ | $2$ | $1$ |
276.48.1-276.o.1.10 | $276$ | $2$ | $2$ | $1$ |
276.48.1-276.p.1.10 | $276$ | $2$ | $2$ | $1$ |
282.48.0-282.a.1.2 | $282$ | $2$ | $2$ | $0$ |
282.48.0-282.b.1.1 | $282$ | $2$ | $2$ | $0$ |
312.48.0-312.fm.1.27 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fn.1.23 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fo.1.31 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fp.1.31 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fq.1.27 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fr.1.27 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.fs.1.31 | $312$ | $2$ | $2$ | $0$ |
312.48.0-312.ft.1.31 | $312$ | $2$ | $2$ | $0$ |
312.48.1-312.hk.1.24 | $312$ | $2$ | $2$ | $1$ |
312.48.1-312.hl.1.24 | $312$ | $2$ | $2$ | $1$ |
312.48.1-312.hm.1.24 | $312$ | $2$ | $2$ | $1$ |
312.48.1-312.hn.1.28 | $312$ | $2$ | $2$ | $1$ |
318.48.0-318.a.1.3 | $318$ | $2$ | $2$ | $0$ |
318.48.0-318.b.1.1 | $318$ | $2$ | $2$ | $0$ |
330.48.0-330.a.1.4 | $330$ | $2$ | $2$ | $0$ |
330.48.0-330.b.1.5 | $330$ | $2$ | $2$ | $0$ |