Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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: | 12.12.1.6 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}4&5\\11&11\end{bmatrix}$, $\begin{bmatrix}11&1\\10&7\end{bmatrix}$, $\begin{bmatrix}11&7\\5&8\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $\SD_{16}\times S_4$ |
| Contains $-I$: | no $\quad$ (see 12.6.1.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: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $4$ | $2$ | $0$ | $0$ | full Jacobian |
| 4.4.0-4.a.1.1 | $4$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.4.0-4.a.1.1 | $4$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.24.1-12.b.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-12.c.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-12.e.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.24.1-12.f.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.1-12.a.1.6 | $12$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 12.48.2-12.b.1.4 | $12$ | $4$ | $4$ | $2$ | $0$ | $1$ |
| 24.24.1-24.g.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.j.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.s.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1-24.v.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 36.36.2-36.a.1.4 | $36$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 36.108.4-36.d.1.2 | $36$ | $9$ | $9$ | $4$ | $0$ | $1^{3}$ |
| 60.24.1-60.b.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-60.c.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-60.e.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1-60.f.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.60.3-60.a.1.2 | $60$ | $5$ | $5$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3-60.a.1.12 | $60$ | $6$ | $6$ | $3$ | $0$ | $1^{2}$ |
| 60.120.5-60.m.1.15 | $60$ | $10$ | $10$ | $5$ | $1$ | $1^{4}$ |
| 84.24.1-84.b.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-84.c.1.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-84.e.1.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.24.1-84.f.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.4-84.a.1.12 | $84$ | $8$ | $8$ | $4$ | $?$ | not computed |
| 84.252.10-84.a.1.16 | $84$ | $21$ | $21$ | $10$ | $?$ | not computed |
| 84.336.13-84.a.1.16 | $84$ | $28$ | $28$ | $13$ | $?$ | not computed |
| 120.24.1-120.g.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.j.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.s.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1-120.v.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.b.1.2 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.c.1.4 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.e.1.3 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.24.1-132.f.1.3 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.144.6-132.a.1.18 | $132$ | $12$ | $12$ | $6$ | $?$ | not computed |
| 156.24.1-156.b.1.2 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-156.c.1.4 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-156.e.1.4 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.24.1-156.f.1.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.168.7-156.a.1.4 | $156$ | $14$ | $14$ | $7$ | $?$ | not computed |
| 168.24.1-168.g.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.j.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.s.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1-168.v.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.b.1.4 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.c.1.4 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.e.1.2 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.24.1-204.f.1.2 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.216.9-204.a.1.11 | $204$ | $18$ | $18$ | $9$ | $?$ | not computed |
| 228.24.1-228.b.1.2 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-228.c.1.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-228.e.1.4 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.24.1-228.f.1.3 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.240.10-228.a.1.9 | $228$ | $20$ | $20$ | $10$ | $?$ | not computed |
| 264.24.1-264.g.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.j.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.s.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1-264.v.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.b.1.3 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.c.1.4 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.e.1.3 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.24.1-276.f.1.3 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.288.12-276.a.1.14 | $276$ | $24$ | $24$ | $12$ | $?$ | not computed |
| 312.24.1-312.g.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.j.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.s.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1-312.v.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |