Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $12$ | $\PSL_2$-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 $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.12.1.12 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&9\\9&4\end{bmatrix}$, $\begin{bmatrix}1&11\\7&10\end{bmatrix}$, $\begin{bmatrix}7&6\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times A_4\times \SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 6.6.1.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $384$ |
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} - 27 $ |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. 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 | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 6 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{y^{2}+27z^{2}}{z^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.1-6.a.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-12.a.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-6.b.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-12.d.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.1-6.a.1.7 | $12$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 12.48.2-12.a.1.3 | $12$ | $4$ | $4$ | $2$ | $0$ | $1$ |
| 24.24.1-24.a.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.d.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.m.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.p.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 36.36.1-18.a.1.7 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.36.1-18.a.1.8 | $36$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 36.36.2-18.a.1.1 | $36$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 36.108.4-18.c.1.1 | $36$ | $9$ | $9$ | $4$ | $0$ | $1^{3}$ |
| 60.24.1-30.a.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-60.a.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-30.b.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-60.d.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.60.3-30.a.1.8 | $60$ | $5$ | $5$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3-30.a.1.8 | $60$ | $6$ | $6$ | $3$ | $0$ | $1^{2}$ |
| 60.120.5-30.e.1.16 | $60$ | $10$ | $10$ | $5$ | $1$ | $1^{4}$ |
| 84.24.1-42.a.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-84.a.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-42.b.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-84.d.1.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.36.1-42.a.1.12 | $84$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 84.36.1-42.a.1.15 | $84$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 84.96.4-42.a.1.12 | $84$ | $8$ | $8$ | $4$ | $?$ | not computed |
| 84.252.10-42.a.1.15 | $84$ | $21$ | $21$ | $10$ | $?$ | not computed |
| 84.336.13-42.a.1.5 | $84$ | $28$ | $28$ | $13$ | $?$ | not computed |
| 120.24.1-120.a.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.d.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.m.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.p.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-66.a.1.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.a.1.2 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-66.b.1.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.d.1.4 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.144.6-66.a.1.6 | $132$ | $12$ | $12$ | $6$ | $?$ | not computed |
| 156.24.1-78.a.1.3 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-156.a.1.2 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-78.b.1.4 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-156.d.1.4 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.36.1-78.a.1.4 | $156$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 156.36.1-78.a.1.11 | $156$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 156.168.7-78.a.1.15 | $156$ | $14$ | $14$ | $7$ | $?$ | not computed |
| 168.24.1-168.a.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.d.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.m.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.p.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-102.a.1.2 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.a.1.2 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-102.b.1.2 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.d.1.4 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.216.9-102.a.1.14 | $204$ | $18$ | $18$ | $9$ | $?$ | not computed |
| 228.24.1-114.a.1.3 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-228.a.1.2 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-114.b.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-228.d.1.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.36.1-114.a.1.15 | $228$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 228.36.1-114.a.1.16 | $228$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 228.240.10-114.a.1.4 | $228$ | $20$ | $20$ | $10$ | $?$ | not computed |
| 252.36.1-126.a.1.4 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.36.1-126.a.1.15 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.36.1-126.b.1.4 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 252.36.1-126.b.1.11 | $252$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.a.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.d.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.m.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.p.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-138.a.1.2 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.a.1.2 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-138.b.1.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.d.1.4 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.288.12-138.a.1.4 | $276$ | $24$ | $24$ | $12$ | $?$ | not computed |
| 312.24.1-312.a.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.d.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.m.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.p.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |