Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $12^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12A3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.3.6 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&0\\0&7\end{bmatrix}$, $\begin{bmatrix}8&3\\9&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times A_4$ |
| Contains $-I$: | no $\quad$ (see 12.48.3.a.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $6$ |
| Cyclic 12-torsion field degree: | $12$ |
| Full 12-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 36.2.a.a, 144.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 3 x^{4} - y^{3} z + z^{4} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{y^{3}(y+2z)^{3}(y^{2}-2yz+4z^{2})^{3}}{z^{3}(y-z)^{3}(y^{2}+yz+z^{2})^{3}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(3)$ | $3$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 4.4.0-2.a.1.1 | $4$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.1-6.a.1.1 | $6$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 12.32.0-12.a.1.1 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.32.0-12.a.2.4 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.48.1-6.a.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.o.1.8 | $12$ | $3$ | $3$ | $7$ | $0$ | $2^{2}$ |
| 12.384.9-12.a.1.1 | $12$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 36.288.7-36.e.1.3 | $36$ | $3$ | $3$ | $7$ | $0$ | $2^{2}$ |
| 36.288.7-36.f.1.4 | $36$ | $3$ | $3$ | $7$ | $4$ | $2^{2}$ |
| 36.288.9-36.y.1.3 | $36$ | $3$ | $3$ | $9$ | $0$ | $2\cdot4$ |
| 36.288.9-36.ba.1.4 | $36$ | $3$ | $3$ | $9$ | $2$ | $1^{2}\cdot2^{2}$ |
| 36.288.9-36.bb.1.4 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{2}\cdot2^{2}$ |
| 36.288.9-36.bc.1.4 | $36$ | $3$ | $3$ | $9$ | $0$ | $2\cdot4$ |
| 36.288.9-36.be.1.4 | $36$ | $3$ | $3$ | $9$ | $1$ | $1^{2}\cdot4$ |
| 36.288.9-36.bg.1.2 | $36$ | $3$ | $3$ | $9$ | $0$ | $2^{3}$ |
| 36.288.9-36.bh.1.4 | $36$ | $3$ | $3$ | $9$ | $0$ | $2^{3}$ |
| 36.288.9-36.bi.1.4 | $36$ | $3$ | $3$ | $9$ | $2$ | $2^{3}$ |
| 36.288.9-36.bj.1.3 | $36$ | $3$ | $3$ | $9$ | $0$ | $2^{3}$ |
| 36.288.9-36.bk.1.4 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 36.288.10-36.e.1.4 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{3}\cdot2^{2}$ |
| 36.288.10-36.e.2.4 | $36$ | $3$ | $3$ | $10$ | $0$ | $1^{3}\cdot2^{2}$ |
| 36.288.11-36.a.1.4 | $36$ | $3$ | $3$ | $11$ | $0$ | $1^{2}\cdot2\cdot4$ |
| 60.480.19-60.a.1.8 | $60$ | $5$ | $5$ | $19$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 60.576.21-60.a.1.16 | $60$ | $6$ | $6$ | $21$ | $0$ | $1^{8}\cdot2\cdot4^{2}$ |
| 60.960.37-60.e.1.16 | $60$ | $10$ | $10$ | $37$ | $3$ | $1^{16}\cdot2^{3}\cdot4^{3}$ |
| 84.288.7-84.fq.1.6 | $84$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 84.288.7-84.fr.1.8 | $84$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 156.288.7-156.ha.1.8 | $156$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 156.288.7-156.hb.1.6 | $156$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 228.288.7-228.fq.1.8 | $228$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 228.288.7-228.fr.1.6 | $228$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.288.7-252.e.1.8 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.288.7-252.f.1.6 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.288.7-252.i.1.6 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.288.7-252.j.1.8 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.288.9-252.bo.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.bq.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.bw.1.7 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.bx.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.by.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.bz.1.6 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.ce.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cg.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.co.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cp.1.7 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cq.1.6 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cr.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cw.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cx.1.7 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cy.1.6 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.cz.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.di.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.dk.1.8 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |