Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.4.29 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&5\\10&5\end{bmatrix}$, $\begin{bmatrix}9&2\\10&3\end{bmatrix}$, $\begin{bmatrix}11&8\\2&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $Q_8:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}$ |
| Newforms: | 24.2.a.a, 36.2.a.a, 48.2.a.a, 72.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x^{2} + 2 z^{2} + 2 z w - w^{2} $ |
| $=$ | $ - x z^{2} - x z w - x w^{2} + 4 y^{3} + 2 y z^{2} - 2 y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} + 6 x^{5} y - 9 x^{4} y^{2} + 2 x^{4} z^{2} - 18 x^{3} y^{3} - 6 x^{3} y z^{2} + \cdots + 3 y^{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 canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\cdot3^4\,\frac{z^{3}(z+2w)^{3}(12xyz^{4}+12xyz^{3}w-12xyzw^{3}-12xyw^{4}-12y^{2}z^{4}+24y^{2}z^{2}w^{2}-12y^{2}w^{4}+5z^{6}+24z^{5}w+45z^{4}w^{2}+32z^{3}w^{3}+3z^{2}w^{4}-w^{6})}{228xyz^{10}+372xyz^{9}w-180xyz^{8}w^{2}-360xyz^{7}w^{3}+432xyz^{6}w^{4}+792xyz^{5}w^{5}-72xyz^{4}w^{6}-1008xyz^{3}w^{7}-612xyz^{2}w^{8}+204xyzw^{9}+204xyw^{10}-216y^{2}z^{10}-432y^{2}z^{9}w+288y^{2}z^{7}w^{3}+216y^{2}z^{6}w^{4}+720y^{2}z^{5}w^{5}+576y^{2}z^{4}w^{6}-576y^{2}z^{3}w^{7}-720y^{2}z^{2}w^{8}+144y^{2}w^{10}+52z^{12}+168z^{11}w+144z^{10}w^{2}-100z^{9}w^{3}-243z^{8}w^{4}+36z^{7}w^{5}+474z^{6}w^{6}+468z^{5}w^{7}+63z^{4}w^{8}-208z^{3}w^{9}-150z^{2}w^{10}+25w^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.2.bn.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.7.bm.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.bo.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.cg.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.cl.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.vy.1 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.144.7.wl.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.bli.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.bmp.1 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 36.216.16.w.1 | $36$ | $3$ | $3$ | $16$ | $4$ | $1^{6}\cdot2^{3}$ |
| 36.648.46.be.1 | $36$ | $9$ | $9$ | $46$ | $17$ | $1^{24}\cdot2^{9}$ |
| 60.144.7.iw.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.iy.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.kn.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.kp.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.360.28.ct.1 | $60$ | $5$ | $5$ | $28$ | $10$ | $1^{24}$ |
| 60.432.31.gp.1 | $60$ | $6$ | $6$ | $31$ | $3$ | $1^{27}$ |
| 60.720.55.xj.1 | $60$ | $10$ | $10$ | $55$ | $22$ | $1^{51}$ |
| 84.144.7.gk.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.gm.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.hm.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.ho.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.ggm.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.ggs.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gwg.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gwm.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.fy.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.ga.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.ha.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.hc.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.hw.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.hy.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.iy.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.ja.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.fiv.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.fjb.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.fvf.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.fvl.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.hk.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.hm.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.im.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.io.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.gk.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.gm.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.hm.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.ho.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fij.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fip.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fut.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fuz.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.fy.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.ga.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.ha.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.hc.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.fsd.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.fsj.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.gen.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.get.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
