Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | Newform level: | $36$ | ||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $1 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $6$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.6.1.2 |
Level structure
| $\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}5&1\\1&4\end{bmatrix}$, $\begin{bmatrix}5&1\\2&5\end{bmatrix}$ |
| $\GL_2(\Z/6\Z)$-subgroup: | $Q_8:S_3$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 6-isogeny field degree: | $12$ |
| Cyclic 6-torsion field degree: | $24$ |
| Full 6-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve has rational points, including 1 rational cusp, 2 rational CM points and 1 known non-cuspidal non-CM point. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(0:-1:1)$, $(0:1:1)$ | |
| no | $\infty$ | $0.000$ | $(0:1:0)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-1:0:1)$ |
| 864.a1 | no | $-13824$ | $= -1 \cdot 2^{9} \cdot 3^{3}$ | $9.534$ | $(2:-3:1)$, $(2:3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 6 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{(y-z)(y+z)}{z^{2}}$ |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 6.2.0.a.1 | $6$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.12.1.c.1 | $6$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 6.12.1.d.1 | $6$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 6.18.1.b.1 | $6$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 12.12.1.g.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.1.j.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.2.c.1 | $12$ | $4$ | $4$ | $2$ | $0$ | $1$ |
| 18.18.2.b.1 | $18$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 18.54.4.e.1 | $18$ | $9$ | $9$ | $4$ | $1$ | $1^{3}$ |
| 24.12.1.y.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.12.1.c.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.12.1.e.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.30.3.b.1 | $30$ | $5$ | $5$ | $3$ | $1$ | $1^{2}$ |
| 30.36.3.c.1 | $30$ | $6$ | $6$ | $3$ | $0$ | $1^{2}$ |
| 30.60.5.h.1 | $30$ | $10$ | $10$ | $5$ | $2$ | $1^{4}$ |
| 42.12.1.c.1 | $42$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 42.12.1.d.1 | $42$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 42.48.4.b.1 | $42$ | $8$ | $8$ | $4$ | $0$ | $1\cdot2$ |
| 42.126.10.c.1 | $42$ | $21$ | $21$ | $10$ | $4$ | $1^{3}\cdot2\cdot4$ |
| 42.168.13.b.1 | $42$ | $28$ | $28$ | $13$ | $4$ | $1^{4}\cdot2^{2}\cdot4$ |
| 60.12.1.g.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.12.1.k.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 66.12.1.c.1 | $66$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 66.12.1.d.1 | $66$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 66.72.6.c.1 | $66$ | $12$ | $12$ | $6$ | $1$ | $1^{5}$ |
| 66.330.26.c.1 | $66$ | $55$ | $55$ | $26$ | $11$ | $1^{3}\cdot2^{5}\cdot4^{3}$ |
| 66.330.26.d.1 | $66$ | $55$ | $55$ | $26$ | $13$ | $1\cdot2^{6}\cdot4^{3}$ |
| 66.396.31.c.1 | $66$ | $66$ | $66$ | $31$ | $14$ | $1^{6}\cdot2^{6}\cdot4^{3}$ |
| 78.12.1.c.1 | $78$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 78.12.1.d.1 | $78$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 78.84.7.b.1 | $78$ | $14$ | $14$ | $7$ | $?$ | not computed |
| 84.12.1.g.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.12.1.j.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 102.12.1.c.1 | $102$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 102.12.1.d.1 | $102$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 102.108.9.c.1 | $102$ | $18$ | $18$ | $9$ | $?$ | not computed |
| 114.12.1.c.1 | $114$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 114.12.1.d.1 | $114$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 114.120.10.b.1 | $114$ | $20$ | $20$ | $10$ | $?$ | not computed |
| 120.12.1.y.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.g.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.j.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 138.12.1.c.1 | $138$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 138.12.1.d.1 | $138$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 138.144.12.c.1 | $138$ | $24$ | $24$ | $12$ | $?$ | not computed |
| 156.12.1.g.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.12.1.j.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.y.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 174.12.1.c.1 | $174$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 174.12.1.d.1 | $174$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 174.180.15.c.1 | $174$ | $30$ | $30$ | $15$ | $?$ | not computed |
| 186.12.1.c.1 | $186$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 186.12.1.d.1 | $186$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 186.192.16.b.1 | $186$ | $32$ | $32$ | $16$ | $?$ | not computed |
| 204.12.1.g.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.12.1.j.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.12.1.c.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.12.1.d.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 222.12.1.c.1 | $222$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 222.12.1.d.1 | $222$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 222.228.19.b.1 | $222$ | $38$ | $38$ | $19$ | $?$ | not computed |
| 228.12.1.g.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.12.1.j.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 246.12.1.c.1 | $246$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 246.12.1.d.1 | $246$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 246.252.21.c.1 | $246$ | $42$ | $42$ | $21$ | $?$ | not computed |
| 258.12.1.c.1 | $258$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 258.12.1.d.1 | $258$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 258.264.22.b.1 | $258$ | $44$ | $44$ | $22$ | $?$ | not computed |
| 264.12.1.y.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.g.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.j.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 282.12.1.c.1 | $282$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 282.12.1.d.1 | $282$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 282.288.24.c.1 | $282$ | $48$ | $48$ | $24$ | $?$ | not computed |
| 312.12.1.y.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 318.12.1.c.1 | $318$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 318.12.1.d.1 | $318$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.12.1.c.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.12.1.d.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |