Invariants
Level: | $3$ | $\SL_2$-level: | $3$ | ||||
Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (none of which are rational) | Cusp widths | $3^{2}$ | Cusp orbits | $2$ | ||
Elliptic points: | $2$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 3C0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 3.6.0.2 |
Sutherland (S) label: | 3Cn |
Level structure
$\GL_2(\Z/3\Z)$-generators: | $\begin{bmatrix}2&1\\2&2\end{bmatrix}$ |
$\GL_2(\Z/3\Z)$-subgroup: | $C_8$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 3-isogeny field degree: | $4$ |
Cyclic 3-torsion field degree: | $8$ |
Full 3-torsion field degree: | $8$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} - x y - 12 x z + y^{2} - 12 y z + 225 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
$X_{\mathrm{ns}}(6)$ | $6$ | $2$ | $2$ | $1$ |
6.12.1.c.1 | $6$ | $2$ | $2$ | $1$ |
6.18.0.a.1 | $6$ | $3$ | $3$ | $0$ |
9.18.0.b.1 | $9$ | $3$ | $3$ | $0$ |
9.18.0.c.1 | $9$ | $3$ | $3$ | $0$ |
$X_{\mathrm{ns}}(9)$ | $9$ | $9$ | $9$ | $2$ |
12.12.1.b.1 | $12$ | $2$ | $2$ | $1$ |
12.12.1.h.1 | $12$ | $2$ | $2$ | $1$ |
12.24.1.m.1 | $12$ | $4$ | $4$ | $1$ |
15.30.2.a.1 | $15$ | $5$ | $5$ | $2$ |
15.36.1.a.1 | $15$ | $6$ | $6$ | $1$ |
15.60.3.c.1 | $15$ | $10$ | $10$ | $3$ |
21.48.3.a.1 | $21$ | $8$ | $8$ | $3$ |
21.126.6.a.1 | $21$ | $21$ | $21$ | $6$ |
21.168.9.e.1 | $21$ | $28$ | $28$ | $9$ |
24.12.1.b.1 | $24$ | $2$ | $2$ | $1$ |
24.12.1.h.1 | $24$ | $2$ | $2$ | $1$ |
24.12.1.z.1 | $24$ | $2$ | $2$ | $1$ |
24.12.1.bf.1 | $24$ | $2$ | $2$ | $1$ |
30.12.1.g.1 | $30$ | $2$ | $2$ | $1$ |
30.12.1.h.1 | $30$ | $2$ | $2$ | $1$ |
33.72.5.a.1 | $33$ | $12$ | $12$ | $5$ |
33.330.20.a.1 | $33$ | $55$ | $55$ | $20$ |
33.330.22.e.1 | $33$ | $55$ | $55$ | $22$ |
33.396.25.a.1 | $33$ | $66$ | $66$ | $25$ |
39.84.5.a.1 | $39$ | $14$ | $14$ | $5$ |
39.468.31.a.1 | $39$ | $78$ | $78$ | $31$ |
39.546.36.a.1 | $39$ | $91$ | $91$ | $36$ |
39.546.38.e.1 | $39$ | $91$ | $91$ | $38$ |
42.12.1.e.1 | $42$ | $2$ | $2$ | $1$ |
42.12.1.f.1 | $42$ | $2$ | $2$ | $1$ |
51.108.7.a.1 | $51$ | $18$ | $18$ | $7$ |
51.816.57.a.1 | $51$ | $136$ | $136$ | $57$ |
51.918.64.a.1 | $51$ | $153$ | $153$ | $64$ |
57.120.9.a.1 | $57$ | $20$ | $20$ | $9$ |
57.1026.72.a.1 | $57$ | $171$ | $171$ | $72$ |
57.1140.81.a.1 | $57$ | $190$ | $190$ | $81$ |
57.1710.126.a.1 | $57$ | $285$ | $285$ | $126$ |
60.12.1.p.1 | $60$ | $2$ | $2$ | $1$ |
60.12.1.s.1 | $60$ | $2$ | $2$ | $1$ |
63.18.0.a.1 | $63$ | $3$ | $3$ | $0$ |
63.18.0.b.1 | $63$ | $3$ | $3$ | $0$ |
63.18.0.c.1 | $63$ | $3$ | $3$ | $0$ |
66.12.1.e.1 | $66$ | $2$ | $2$ | $1$ |
66.12.1.f.1 | $66$ | $2$ | $2$ | $1$ |
69.144.11.a.1 | $69$ | $24$ | $24$ | $11$ |
69.1518.110.a.1 | $69$ | $253$ | $253$ | $110$ |
69.1656.121.a.1 | $69$ | $276$ | $276$ | $121$ |
78.12.1.e.1 | $78$ | $2$ | $2$ | $1$ |
78.12.1.f.1 | $78$ | $2$ | $2$ | $1$ |
84.12.1.n.1 | $84$ | $2$ | $2$ | $1$ |
84.12.1.q.1 | $84$ | $2$ | $2$ | $1$ |
87.180.13.a.1 | $87$ | $30$ | $30$ | $13$ |
93.192.15.a.1 | $93$ | $32$ | $32$ | $15$ |
102.12.1.e.1 | $102$ | $2$ | $2$ | $1$ |
102.12.1.f.1 | $102$ | $2$ | $2$ | $1$ |
111.228.17.a.1 | $111$ | $38$ | $38$ | $17$ |
114.12.1.e.1 | $114$ | $2$ | $2$ | $1$ |
114.12.1.f.1 | $114$ | $2$ | $2$ | $1$ |
117.18.0.a.1 | $117$ | $3$ | $3$ | $0$ |
117.18.0.b.1 | $117$ | $3$ | $3$ | $0$ |
117.18.0.c.1 | $117$ | $3$ | $3$ | $0$ |
120.12.1.cb.1 | $120$ | $2$ | $2$ | $1$ |
120.12.1.ch.1 | $120$ | $2$ | $2$ | $1$ |
120.12.1.cn.1 | $120$ | $2$ | $2$ | $1$ |
120.12.1.ct.1 | $120$ | $2$ | $2$ | $1$ |
123.252.19.a.1 | $123$ | $42$ | $42$ | $19$ |
129.264.21.a.1 | $129$ | $44$ | $44$ | $21$ |
132.12.1.n.1 | $132$ | $2$ | $2$ | $1$ |
132.12.1.q.1 | $132$ | $2$ | $2$ | $1$ |
138.12.1.e.1 | $138$ | $2$ | $2$ | $1$ |
138.12.1.f.1 | $138$ | $2$ | $2$ | $1$ |
141.288.23.a.1 | $141$ | $48$ | $48$ | $23$ |
156.12.1.n.1 | $156$ | $2$ | $2$ | $1$ |
156.12.1.q.1 | $156$ | $2$ | $2$ | $1$ |
168.12.1.bx.1 | $168$ | $2$ | $2$ | $1$ |
168.12.1.cd.1 | $168$ | $2$ | $2$ | $1$ |
168.12.1.cj.1 | $168$ | $2$ | $2$ | $1$ |
168.12.1.cp.1 | $168$ | $2$ | $2$ | $1$ |
171.18.0.a.1 | $171$ | $3$ | $3$ | $0$ |
171.18.0.b.1 | $171$ | $3$ | $3$ | $0$ |
171.18.0.c.1 | $171$ | $3$ | $3$ | $0$ |
174.12.1.e.1 | $174$ | $2$ | $2$ | $1$ |
174.12.1.f.1 | $174$ | $2$ | $2$ | $1$ |
186.12.1.e.1 | $186$ | $2$ | $2$ | $1$ |
186.12.1.f.1 | $186$ | $2$ | $2$ | $1$ |
204.12.1.n.1 | $204$ | $2$ | $2$ | $1$ |
204.12.1.q.1 | $204$ | $2$ | $2$ | $1$ |
210.12.1.s.1 | $210$ | $2$ | $2$ | $1$ |
210.12.1.t.1 | $210$ | $2$ | $2$ | $1$ |
222.12.1.e.1 | $222$ | $2$ | $2$ | $1$ |
222.12.1.f.1 | $222$ | $2$ | $2$ | $1$ |
228.12.1.n.1 | $228$ | $2$ | $2$ | $1$ |
228.12.1.q.1 | $228$ | $2$ | $2$ | $1$ |
246.12.1.e.1 | $246$ | $2$ | $2$ | $1$ |
246.12.1.f.1 | $246$ | $2$ | $2$ | $1$ |
258.12.1.e.1 | $258$ | $2$ | $2$ | $1$ |
258.12.1.f.1 | $258$ | $2$ | $2$ | $1$ |
264.12.1.bx.1 | $264$ | $2$ | $2$ | $1$ |
264.12.1.cd.1 | $264$ | $2$ | $2$ | $1$ |
264.12.1.cj.1 | $264$ | $2$ | $2$ | $1$ |
264.12.1.cp.1 | $264$ | $2$ | $2$ | $1$ |
276.12.1.n.1 | $276$ | $2$ | $2$ | $1$ |
276.12.1.q.1 | $276$ | $2$ | $2$ | $1$ |
279.18.0.a.1 | $279$ | $3$ | $3$ | $0$ |
279.18.0.b.1 | $279$ | $3$ | $3$ | $0$ |
279.18.0.c.1 | $279$ | $3$ | $3$ | $0$ |
282.12.1.e.1 | $282$ | $2$ | $2$ | $1$ |
282.12.1.f.1 | $282$ | $2$ | $2$ | $1$ |
312.12.1.bx.1 | $312$ | $2$ | $2$ | $1$ |
312.12.1.cd.1 | $312$ | $2$ | $2$ | $1$ |
312.12.1.cj.1 | $312$ | $2$ | $2$ | $1$ |
312.12.1.cp.1 | $312$ | $2$ | $2$ | $1$ |
318.12.1.e.1 | $318$ | $2$ | $2$ | $1$ |
318.12.1.f.1 | $318$ | $2$ | $2$ | $1$ |
330.12.1.s.1 | $330$ | $2$ | $2$ | $1$ |
330.12.1.t.1 | $330$ | $2$ | $2$ | $1$ |
333.18.0.a.1 | $333$ | $3$ | $3$ | $0$ |
333.18.0.b.1 | $333$ | $3$ | $3$ | $0$ |
333.18.0.c.1 | $333$ | $3$ | $3$ | $0$ |