Invariants
| Level: | $3$ | $\SL_2$-level: | $3$ | ||||
| Index: | $4$ | $\PSL_2$-index: | $4$ | ||||
| Genus: | $0 = 1 + \frac{ 4 }{12} - \frac{ 0 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot3$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 3B0 |
| Sutherland and Zywina (SZ) label: | 3B0-3a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 3.4.0.1 |
| Sutherland (S) label: | 3B |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 78278 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 4 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{x^{4}(x-18y)^{3}(x+30y)}{y^{3}x^{4}(x-24y)}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $3$ | $3$ | $0$ |
| 6.8.0.a.1 | $6$ | $2$ | $2$ | $0$ |
| 6.8.0.b.1 | $6$ | $2$ | $2$ | $0$ |
| $X_0(6)$ | $6$ | $3$ | $3$ | $0$ |
| $X_0(9)$ | $9$ | $3$ | $3$ | $0$ |
| 9.12.0.b.1 | $9$ | $3$ | $3$ | $0$ |
| 9.12.1.a.1 | $9$ | $3$ | $3$ | $1$ |
| 12.8.0.a.1 | $12$ | $2$ | $2$ | $0$ |
| 12.8.0.b.1 | $12$ | $2$ | $2$ | $0$ |
| 12.16.1.a.1 | $12$ | $4$ | $4$ | $1$ |
| 15.20.1.a.1 | $15$ | $5$ | $5$ | $1$ |
| $X_0(15)$ | $15$ | $6$ | $6$ | $1$ |
| 15.40.2.a.1 | $15$ | $10$ | $10$ | $2$ |
| $X_0(21)$ | $21$ | $8$ | $8$ | $1$ |
| 21.84.5.a.1 | $21$ | $21$ | $21$ | $5$ |
| 21.112.6.a.1 | $21$ | $28$ | $28$ | $6$ |
| 24.8.0.a.1 | $24$ | $2$ | $2$ | $0$ |
| 24.8.0.b.1 | $24$ | $2$ | $2$ | $0$ |
| 24.8.0.c.1 | $24$ | $2$ | $2$ | $0$ |
| 24.8.0.d.1 | $24$ | $2$ | $2$ | $0$ |
| 30.8.0.a.1 | $30$ | $2$ | $2$ | $0$ |
| 30.8.0.b.1 | $30$ | $2$ | $2$ | $0$ |
| $X_0(33)$ | $33$ | $12$ | $12$ | $3$ |
| 33.220.13.a.1 | $33$ | $55$ | $55$ | $13$ |
| 33.220.14.a.1 | $33$ | $55$ | $55$ | $14$ |
| 33.264.17.a.1 | $33$ | $66$ | $66$ | $17$ |
| $X_0(39)$ | $39$ | $14$ | $14$ | $3$ |
| 39.312.21.a.1 | $39$ | $78$ | $78$ | $21$ |
| 39.364.23.a.1 | $39$ | $91$ | $91$ | $23$ |
| 39.364.24.a.1 | $39$ | $91$ | $91$ | $24$ |
| 42.8.0.a.1 | $42$ | $2$ | $2$ | $0$ |
| 42.8.0.b.1 | $42$ | $2$ | $2$ | $0$ |
| $X_0(51)$ | $51$ | $18$ | $18$ | $5$ |
| 51.544.38.a.1 | $51$ | $136$ | $136$ | $38$ |
| 51.612.43.a.1 | $51$ | $153$ | $153$ | $43$ |
| $X_0(57)$ | $57$ | $20$ | $20$ | $5$ |
| 57.684.49.a.1 | $57$ | $171$ | $171$ | $49$ |
| 57.760.54.a.1 | $57$ | $190$ | $190$ | $54$ |
| 57.1140.79.a.1 | $57$ | $285$ | $285$ | $79$ |
| 60.8.0.a.1 | $60$ | $2$ | $2$ | $0$ |
| 60.8.0.b.1 | $60$ | $2$ | $2$ | $0$ |
| 66.8.0.a.1 | $66$ | $2$ | $2$ | $0$ |
| 66.8.0.b.1 | $66$ | $2$ | $2$ | $0$ |
| $X_0(69)$ | $69$ | $24$ | $24$ | $7$ |
| 69.1012.74.a.1 | $69$ | $253$ | $253$ | $74$ |
| 69.1104.81.a.1 | $69$ | $276$ | $276$ | $81$ |
| 78.8.0.a.1 | $78$ | $2$ | $2$ | $0$ |
| 78.8.0.b.1 | $78$ | $2$ | $2$ | $0$ |
| 84.8.0.a.1 | $84$ | $2$ | $2$ | $0$ |
| 84.8.0.b.1 | $84$ | $2$ | $2$ | $0$ |
| $X_0(87)$ | $87$ | $30$ | $30$ | $9$ |
| $X_0(93)$ | $93$ | $32$ | $32$ | $9$ |
| 102.8.0.a.1 | $102$ | $2$ | $2$ | $0$ |
| 102.8.0.b.1 | $102$ | $2$ | $2$ | $0$ |
| $X_0(111)$ | $111$ | $38$ | $38$ | $11$ |
| 114.8.0.a.1 | $114$ | $2$ | $2$ | $0$ |
| 114.8.0.b.1 | $114$ | $2$ | $2$ | $0$ |
| 120.8.0.a.1 | $120$ | $2$ | $2$ | $0$ |
| 120.8.0.b.1 | $120$ | $2$ | $2$ | $0$ |
| 120.8.0.c.1 | $120$ | $2$ | $2$ | $0$ |
| 120.8.0.d.1 | $120$ | $2$ | $2$ | $0$ |
| $X_0(123)$ | $123$ | $42$ | $42$ | $13$ |
| $X_0(129)$ | $129$ | $44$ | $44$ | $13$ |
| 132.8.0.a.1 | $132$ | $2$ | $2$ | $0$ |
| 132.8.0.b.1 | $132$ | $2$ | $2$ | $0$ |
| 138.8.0.a.1 | $138$ | $2$ | $2$ | $0$ |
| 138.8.0.b.1 | $138$ | $2$ | $2$ | $0$ |
| $X_0(141)$ | $141$ | $48$ | $48$ | $15$ |
| 156.8.0.a.1 | $156$ | $2$ | $2$ | $0$ |
| 156.8.0.b.1 | $156$ | $2$ | $2$ | $0$ |
| $X_0(159)$ | $159$ | $54$ | $54$ | $17$ |
| 168.8.0.a.1 | $168$ | $2$ | $2$ | $0$ |
| 168.8.0.b.1 | $168$ | $2$ | $2$ | $0$ |
| 168.8.0.c.1 | $168$ | $2$ | $2$ | $0$ |
| 168.8.0.d.1 | $168$ | $2$ | $2$ | $0$ |
| 174.8.0.a.1 | $174$ | $2$ | $2$ | $0$ |
| 174.8.0.b.1 | $174$ | $2$ | $2$ | $0$ |
| $X_0(177)$ | $177$ | $60$ | $60$ | $19$ |
| $X_0(183)$ | $183$ | $62$ | $62$ | $19$ |
| 186.8.0.a.1 | $186$ | $2$ | $2$ | $0$ |
| 186.8.0.b.1 | $186$ | $2$ | $2$ | $0$ |
| $X_0(201)$ | $201$ | $68$ | $68$ | $21$ |
| 204.8.0.a.1 | $204$ | $2$ | $2$ | $0$ |
| 204.8.0.b.1 | $204$ | $2$ | $2$ | $0$ |
| 210.8.0.a.1 | $210$ | $2$ | $2$ | $0$ |
| 210.8.0.b.1 | $210$ | $2$ | $2$ | $0$ |
| $X_0(213)$ | $213$ | $72$ | $72$ | $23$ |
| $X_0(219)$ | $219$ | $74$ | $74$ | $23$ |
| 222.8.0.a.1 | $222$ | $2$ | $2$ | $0$ |
| 222.8.0.b.1 | $222$ | $2$ | $2$ | $0$ |
| 228.8.0.a.1 | $228$ | $2$ | $2$ | $0$ |
| 228.8.0.b.1 | $228$ | $2$ | $2$ | $0$ |
| 246.8.0.a.1 | $246$ | $2$ | $2$ | $0$ |
| 246.8.0.b.1 | $246$ | $2$ | $2$ | $0$ |
| 258.8.0.a.1 | $258$ | $2$ | $2$ | $0$ |
| 258.8.0.b.1 | $258$ | $2$ | $2$ | $0$ |
| 264.8.0.a.1 | $264$ | $2$ | $2$ | $0$ |
| 264.8.0.b.1 | $264$ | $2$ | $2$ | $0$ |
| 264.8.0.c.1 | $264$ | $2$ | $2$ | $0$ |
| 264.8.0.d.1 | $264$ | $2$ | $2$ | $0$ |
| 276.8.0.a.1 | $276$ | $2$ | $2$ | $0$ |
| 276.8.0.b.1 | $276$ | $2$ | $2$ | $0$ |
| 282.8.0.a.1 | $282$ | $2$ | $2$ | $0$ |
| 282.8.0.b.1 | $282$ | $2$ | $2$ | $0$ |
| 312.8.0.a.1 | $312$ | $2$ | $2$ | $0$ |
| 312.8.0.b.1 | $312$ | $2$ | $2$ | $0$ |
| 312.8.0.c.1 | $312$ | $2$ | $2$ | $0$ |
| 312.8.0.d.1 | $312$ | $2$ | $2$ | $0$ |
| 318.8.0.a.1 | $318$ | $2$ | $2$ | $0$ |
| 318.8.0.b.1 | $318$ | $2$ | $2$ | $0$ |
| 330.8.0.a.1 | $330$ | $2$ | $2$ | $0$ |
| 330.8.0.b.1 | $330$ | $2$ | $2$ | $0$ |