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: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.4.5 |
Level structure
Jacobian
| Conductor: | $2^{14}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a, 144.2.a.a$^{3}$ |
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 $\Q_p$ points for $p=7$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.24.0.a.1 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.36.1.bu.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
| 12.36.2.a.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 12.36.2.c.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.i.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.j.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.p.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.q.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.bg.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.bl.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 24.144.7.cx.1 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.144.7.dc.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.9.q.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.s.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.fc.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.144.9.fd.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.144.9.hg.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.hh.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.iu.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.144.9.iw.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 36.216.16.c.1 | $36$ | $3$ | $3$ | $16$ | $3$ | $1^{12}$ |
| 36.648.46.h.1 | $36$ | $9$ | $9$ | $46$ | $15$ | $1^{36}\cdot2^{3}$ |
| 60.144.7.i.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 60.144.7.j.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.s.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 60.144.7.t.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.360.28.c.1 | $60$ | $5$ | $5$ | $28$ | $12$ | $1^{24}$ |
| 60.432.31.c.1 | $60$ | $6$ | $6$ | $31$ | $6$ | $1^{27}$ |
| 60.720.55.bg.1 | $60$ | $10$ | $10$ | $55$ | $23$ | $1^{51}$ |
| 84.144.7.i.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.j.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.p.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.q.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.bg.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.bl.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.cw.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.db.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.9.fo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.fp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.gm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.gn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.iu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.iv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ki.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.kj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.144.7.i.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.j.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.p.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.q.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.i.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.j.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.p.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.q.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.bg.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.bl.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.cp.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.cu.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.9.fo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.fp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.gm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.gn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.iq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.ir.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.ke.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.144.7.i.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.j.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.p.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.q.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.i.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.j.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.p.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.q.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.bg.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.bl.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.cp.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.cu.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.9.fo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.fp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.gm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.gn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.iq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.ir.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.ke.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.144.7.i.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.j.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.p.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.q.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.bg.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.bl.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.cp.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.cu.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.9.fo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.fp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.gm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.gn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.iq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.ir.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.ke.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |