Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.1.45 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&2\\4&9\end{bmatrix}$, $\begin{bmatrix}9&7\\4&9\end{bmatrix}$, $\begin{bmatrix}11&0\\6&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^3:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y - y^{2} - z^{2} $ |
| $=$ | $3 x^{2} - 8 x y - 4 y^{2} - 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 3 x^{2} y^{2} + 2 x^{2} z^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^6}\cdot\frac{(432z^{6}-w^{6})^{3}}{w^{6}z^{12}}$ |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.0.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.36.0.d.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.36.1.bh.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.5.h.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.l.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.q.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.w.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bx.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.df.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.fu.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.hj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 36.216.9.c.1 | $36$ | $3$ | $3$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 36.216.9.f.1 | $36$ | $3$ | $3$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 36.216.9.r.1 | $36$ | $3$ | $3$ | $9$ | $6$ | $1^{4}\cdot2^{2}$ |
| 60.144.5.qq.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.qs.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.qy.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.ra.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.360.25.cft.1 | $60$ | $5$ | $5$ | $25$ | $12$ | $1^{24}$ |
| 60.432.25.bkn.1 | $60$ | $6$ | $6$ | $25$ | $4$ | $1^{24}$ |
| 60.720.49.ekd.1 | $60$ | $10$ | $10$ | $49$ | $24$ | $1^{48}$ |
| 84.144.5.hm.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.ho.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.hu.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.hw.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ese.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ess.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eui.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.euw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.hm.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.ho.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.hu.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.hw.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.hm.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.ho.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.hu.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.hw.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cee.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ces.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cgi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cgw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.hm.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.ho.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.hu.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.hw.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.hm.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.ho.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.hu.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.hw.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cee.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ces.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cgi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cgw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.hm.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.ho.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.hu.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.hw.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cee.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ces.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cgi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cgw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |