Invariants
| Level: | $12$ | $\SL_2$-level: | $6$ | Newform level: | $144$ | ||
| 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 $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.6.1.2 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&6\\9&1\end{bmatrix}$, $\begin{bmatrix}6&1\\7&0\end{bmatrix}$, $\begin{bmatrix}11&5\\5&10\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $Q_8:\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $768$ |
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} - 1 $ |
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 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 6 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{y^{2}+z^{2}}{z^{2}}$ |
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 |
| 12.2.0.a.1 | $12$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.12.1.h.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.1.i.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.1.k.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.12.1.l.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.18.1.d.1 | $12$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 12.24.2.d.1 | $12$ | $4$ | $4$ | $2$ | $0$ | $1$ |
| 24.12.1.be.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.12.1.bt.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 36.18.2.b.1 | $36$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 36.54.4.h.1 | $36$ | $9$ | $9$ | $4$ | $1$ | $1^{3}$ |
| 60.12.1.h.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.12.1.i.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.12.1.l.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.12.1.m.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.30.3.b.1 | $60$ | $5$ | $5$ | $3$ | $1$ | $1^{2}$ |
| 60.36.3.b.1 | $60$ | $6$ | $6$ | $3$ | $0$ | $1^{2}$ |
| 60.60.5.t.1 | $60$ | $10$ | $10$ | $5$ | $2$ | $1^{4}$ |
| 84.12.1.h.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.12.1.i.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.12.1.k.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.12.1.l.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.48.4.b.1 | $84$ | $8$ | $8$ | $4$ | $?$ | not computed |
| 84.126.10.b.1 | $84$ | $21$ | $21$ | $10$ | $?$ | not computed |
| 84.168.13.b.1 | $84$ | $28$ | $28$ | $13$ | $?$ | not computed |
| 120.12.1.be.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.12.1.bv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.h.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.i.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.k.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.12.1.l.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.72.6.b.1 | $132$ | $12$ | $12$ | $6$ | $?$ | not computed |
| 156.12.1.h.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.12.1.i.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.12.1.k.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.12.1.l.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.84.7.b.1 | $156$ | $14$ | $14$ | $7$ | $?$ | not computed |
| 168.12.1.be.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.12.1.bt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.12.1.h.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.12.1.i.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.12.1.k.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.12.1.l.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.108.9.b.1 | $204$ | $18$ | $18$ | $9$ | $?$ | not computed |
| 228.12.1.h.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.12.1.i.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.12.1.k.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.12.1.l.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.120.10.b.1 | $228$ | $20$ | $20$ | $10$ | $?$ | not computed |
| 264.12.1.be.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.12.1.bt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.h.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.i.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.k.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.12.1.l.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.144.12.b.1 | $276$ | $24$ | $24$ | $12$ | $?$ | not computed |
| 312.12.1.be.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.12.1.bt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |