Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12G3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.144.3.166 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}9&4\\10&3\end{bmatrix}$, $\begin{bmatrix}9&10\\8&9\end{bmatrix}$, $\begin{bmatrix}11&2\\10&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times \SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 12.72.3.h.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: | $2^{8}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 36.2.a.a$^{2}$, 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 x z - 2 x u - y w - w^{2} - 2 w t $ |
| $=$ | $3 y^{2} - z^{2} - z u - u^{2}$ | |
| $=$ | $3 x^{2} + x z - x u + y^{2} + y t - z u - w^{2} + t^{2}$ | |
| $=$ | $z^{2} - z u - 3 w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} - 2 x^{6} y^{2} - 4 x^{6} y z - 14 x^{6} z^{2} + x^{4} y^{4} + 4 x^{4} y^{3} z + 6 x^{4} y^{2} z^{2} + \cdots + 144 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 3 w^{2} $ | $=$ | $ 9 x^{4} + 3 x^{2} z^{2} + z^{4} $ |
| $0$ | $=$ | $-3 x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{5184zw^{8}+17280zw^{6}u^{2}+7344zw^{4}u^{4}+804zw^{2}u^{6}+28zu^{8}+20736w^{8}u+15984w^{6}u^{3}+2160w^{4}u^{5}+84w^{2}u^{7}-u^{9}}{u^{6}(3zw^{2}+zu^{2}+3w^{2}u)}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.72.3.h.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}-2X^{6}Y^{2}+X^{4}Y^{4}-4X^{6}YZ+4X^{4}Y^{3}Z-14X^{6}Z^{2}+6X^{4}Y^{2}Z^{2}+4X^{4}YZ^{3}+61X^{4}Z^{4}+12X^{2}Y^{2}Z^{4}+24X^{2}YZ^{5}-132X^{2}Z^{6}+144Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.1-12.a.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 12.72.1-12.a.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 12.72.2-12.d.1.2 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.72.2-12.f.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.72.2-12.f.1.4 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.m.1.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 12.288.7-12.m.1.2 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 12.288.7-12.n.1.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 12.288.7-12.n.1.3 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 24.288.7-24.ce.1.4 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.ce.1.7 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 24.288.7-24.ck.1.2 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 24.288.7-24.ck.1.5 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 24.288.7-24.cq.1.3 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 24.288.7-24.cr.1.3 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 24.288.9-24.jy.1.3 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 24.288.9-24.jz.1.3 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 36.432.15-36.g.1.3 | $36$ | $3$ | $3$ | $15$ | $2$ | $1^{12}$ |
| 36.1296.43-36.h.1.1 | $36$ | $9$ | $9$ | $43$ | $20$ | $1^{24}\cdot2^{8}$ |
| 60.288.7-60.bx.1.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 60.288.7-60.bx.1.7 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 60.288.7-60.by.1.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 60.288.7-60.by.1.5 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 60.720.27-60.h.1.4 | $60$ | $5$ | $5$ | $27$ | $13$ | $1^{24}$ |
| 60.864.29-60.ct.1.8 | $60$ | $6$ | $6$ | $29$ | $5$ | $1^{26}$ |
| 60.1440.53-60.gt.1.1 | $60$ | $10$ | $10$ | $53$ | $27$ | $1^{50}$ |
| 84.288.7-84.bn.1.2 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.bn.1.8 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.bo.1.2 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.bo.1.6 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.je.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.je.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.jl.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.jl.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.jt.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ju.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.9-120.bcg.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bch.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.288.7-132.bn.1.2 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.bn.1.8 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.bo.1.2 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.288.7-132.bo.1.6 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.bn.1.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.bn.1.7 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.bo.1.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.bo.1.5 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.ie.1.4 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.ie.1.14 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.il.1.2 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.il.1.7 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.it.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.iu.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.9-168.bbu.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.bbv.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.288.7-204.bn.1.2 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.bn.1.3 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.bo.1.5 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.288.7-204.bo.1.7 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.bn.1.2 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.bn.1.8 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.bo.1.2 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.288.7-228.bo.1.6 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.ie.1.4 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.ie.1.15 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.il.1.2 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.il.1.11 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.it.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.iu.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.9-264.bby.1.16 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.288.9-264.bbz.1.16 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.288.7-276.bn.1.2 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.bn.1.8 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.bo.1.2 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.288.7-276.bo.1.6 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.ie.1.3 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.ie.1.13 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.il.1.3 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.il.1.13 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.it.1.7 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.iu.1.7 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.9-312.bbu.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.288.9-312.bbv.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |