Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-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: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.144.4.9 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&0\\6&7\end{bmatrix}$, $\begin{bmatrix}7&2\\6&5\end{bmatrix}$, $\begin{bmatrix}7&10\\10&9\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times \SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 12.72.4.d.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $32$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 3 x^{2} + z^{2} + w^{2} $ |
| $=$ | $ - x z w + 12 y^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} z^{2} + 9 x^{2} z^{4} + 16 y^{6} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 12.72.4.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}Z^{2}+9X^{2}Z^{4}+16Y^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.b.1.2 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.72.2-12.a.1.2 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 12.72.2-12.a.1.8 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 12.72.2-12.d.1.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 12.72.2-12.d.1.8 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.288.7-12.k.1.2 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.288.7-12.m.1.2 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.288.7-12.r.1.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.288.7-12.t.1.2 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.288.7-24.bq.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.288.7-24.cc.1.4 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 24.288.7-24.dh.1.4 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.288.7-24.dt.1.4 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.288.9-24.cj.1.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.288.9-24.ck.1.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.288.9-24.ga.1.2 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.288.9-24.gb.1.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.288.9-24.ie.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.288.9-24.if.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.288.9-24.iz.1.5 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.288.9-24.ja.1.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.432.16-36.d.1.3 | $36$ | $3$ | $3$ | $16$ | $3$ | $1^{12}$ |
| 36.1296.46-36.i.1.2 | $36$ | $9$ | $9$ | $46$ | $15$ | $1^{36}\cdot2^{3}$ |
| 60.288.7-60.k.1.4 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.288.7-60.m.1.4 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
| 60.288.7-60.u.1.4 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
| 60.288.7-60.w.1.3 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.720.28-60.d.1.4 | $60$ | $5$ | $5$ | $28$ | $10$ | $1^{24}$ |
| 60.864.31-60.d.1.12 | $60$ | $6$ | $6$ | $31$ | $4$ | $1^{27}$ |
| 60.1440.55-60.bh.1.3 | $60$ | $10$ | $10$ | $55$ | $21$ | $1^{51}$ |
| 84.288.7-84.k.1.4 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.m.1.4 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.r.1.3 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.t.1.3 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bq.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cc.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dg.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ds.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.9-120.ga.1.13 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.gb.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.hk.1.8 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.hl.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.js.1.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.jt.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ku.1.13 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.kv.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.288.7-132.k.1.4 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.m.1.4 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.r.1.3 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.t.1.4 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.k.1.4 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.m.1.4 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.r.1.3 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.t.1.3 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bq.1.8 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cc.1.8 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cz.1.8 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.dl.1.8 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.9-168.ga.1.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.gb.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.hk.1.8 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.hl.1.6 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.jo.1.4 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.jp.1.2 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.kq.1.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.kr.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.288.7-204.k.1.4 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.m.1.4 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.r.1.4 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.t.1.4 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.k.1.4 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.m.1.4 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.r.1.2 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.t.1.4 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.bq.1.8 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cc.1.8 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cz.1.8 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.dl.1.8 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.9-264.ga.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.gb.1.15 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.hk.1.8 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.hl.1.6 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.jo.1.4 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.jp.1.2 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.kq.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.kr.1.15 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.288.7-276.k.1.4 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.m.1.4 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.r.1.2 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.t.1.4 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.bq.1.8 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cc.1.8 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cz.1.8 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.dl.1.8 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.9-312.ga.1.13 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.gb.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.hk.1.8 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.hl.1.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.jo.1.4 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.jp.1.2 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.kq.1.13 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.kr.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |