Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $12^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-19,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 12F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.0.39 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}2&9\\7&10\end{bmatrix}$, $\begin{bmatrix}3&8\\8&7\end{bmatrix}$, $\begin{bmatrix}7&5\\11&2\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_{24}:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $192$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^{18}\cdot3^3\,\frac{(3x-y)^{3}(12x-y)^{27}(432x^{3}-216x^{2}y+36xy^{2}-y^{3})^{3}(864x^{3}-432x^{2}y+72xy^{2}-5y^{3})^{3}}{(12x-y)^{24}(72x^{2}-12xy-y^{2})^{12}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(6)$ | $6$ | $4$ | $4$ | $0$ | $0$ |
| 12.12.0.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{ns}}(12)$ | $12$ | $2$ | $2$ | $3$ |
| 12.48.3.f.1 | $12$ | $2$ | $2$ | $3$ |
| 12.48.3.r.1 | $12$ | $2$ | $2$ | $3$ |
| 12.48.3.t.1 | $12$ | $2$ | $2$ | $3$ |
| 12.72.2.o.1 | $12$ | $3$ | $3$ | $2$ |
| 24.48.3.i.1 | $24$ | $2$ | $2$ | $3$ |
| 24.48.3.r.1 | $24$ | $2$ | $2$ | $3$ |
| 24.48.3.cb.1 | $24$ | $2$ | $2$ | $3$ |
| 24.48.3.ch.1 | $24$ | $2$ | $2$ | $3$ |
| $X_{\mathrm{ns}}^+(24)$ | $24$ | $4$ | $4$ | $3$ |
| 36.72.4.w.1 | $36$ | $3$ | $3$ | $4$ |
| 36.72.4.x.1 | $36$ | $3$ | $3$ | $4$ |
| $X_{\mathrm{ns}}^+(36)$ | $36$ | $9$ | $9$ | $10$ |
| 60.48.3.ch.1 | $60$ | $2$ | $2$ | $3$ |
| 60.48.3.cj.1 | $60$ | $2$ | $2$ | $3$ |
| 60.48.3.cl.1 | $60$ | $2$ | $2$ | $3$ |
| 60.48.3.cn.1 | $60$ | $2$ | $2$ | $3$ |
| 60.120.8.cp.1 | $60$ | $5$ | $5$ | $8$ |
| 60.144.7.un.1 | $60$ | $6$ | $6$ | $7$ |
| 60.240.15.kt.1 | $60$ | $10$ | $10$ | $15$ |
| 84.48.3.bt.1 | $84$ | $2$ | $2$ | $3$ |
| 84.48.3.bv.1 | $84$ | $2$ | $2$ | $3$ |
| 84.48.3.bx.1 | $84$ | $2$ | $2$ | $3$ |
| 84.48.3.bz.1 | $84$ | $2$ | $2$ | $3$ |
| 84.192.15.ch.1 | $84$ | $8$ | $8$ | $15$ |
| 120.48.3.gj.1 | $120$ | $2$ | $2$ | $3$ |
| 120.48.3.gp.1 | $120$ | $2$ | $2$ | $3$ |
| 120.48.3.gv.1 | $120$ | $2$ | $2$ | $3$ |
| 120.48.3.hb.1 | $120$ | $2$ | $2$ | $3$ |
| 132.48.3.bt.1 | $132$ | $2$ | $2$ | $3$ |
| 132.48.3.bv.1 | $132$ | $2$ | $2$ | $3$ |
| 132.48.3.bx.1 | $132$ | $2$ | $2$ | $3$ |
| 132.48.3.bz.1 | $132$ | $2$ | $2$ | $3$ |
| 132.288.23.cj.1 | $132$ | $12$ | $12$ | $23$ |
| 156.48.3.bt.1 | $156$ | $2$ | $2$ | $3$ |
| 156.48.3.bv.1 | $156$ | $2$ | $2$ | $3$ |
| 156.48.3.bx.1 | $156$ | $2$ | $2$ | $3$ |
| 156.48.3.bz.1 | $156$ | $2$ | $2$ | $3$ |
| 156.336.23.gh.1 | $156$ | $14$ | $14$ | $23$ |
| 168.48.3.fp.1 | $168$ | $2$ | $2$ | $3$ |
| 168.48.3.fv.1 | $168$ | $2$ | $2$ | $3$ |
| 168.48.3.gb.1 | $168$ | $2$ | $2$ | $3$ |
| 168.48.3.gh.1 | $168$ | $2$ | $2$ | $3$ |
| 204.48.3.bt.1 | $204$ | $2$ | $2$ | $3$ |
| 204.48.3.bv.1 | $204$ | $2$ | $2$ | $3$ |
| 204.48.3.bx.1 | $204$ | $2$ | $2$ | $3$ |
| 204.48.3.bz.1 | $204$ | $2$ | $2$ | $3$ |
| 228.48.3.bt.1 | $228$ | $2$ | $2$ | $3$ |
| 228.48.3.bv.1 | $228$ | $2$ | $2$ | $3$ |
| 228.48.3.bx.1 | $228$ | $2$ | $2$ | $3$ |
| 228.48.3.bz.1 | $228$ | $2$ | $2$ | $3$ |
| 264.48.3.fp.1 | $264$ | $2$ | $2$ | $3$ |
| 264.48.3.fv.1 | $264$ | $2$ | $2$ | $3$ |
| 264.48.3.gb.1 | $264$ | $2$ | $2$ | $3$ |
| 264.48.3.gh.1 | $264$ | $2$ | $2$ | $3$ |
| 276.48.3.bt.1 | $276$ | $2$ | $2$ | $3$ |
| 276.48.3.bv.1 | $276$ | $2$ | $2$ | $3$ |
| 276.48.3.bx.1 | $276$ | $2$ | $2$ | $3$ |
| 276.48.3.bz.1 | $276$ | $2$ | $2$ | $3$ |
| 312.48.3.fp.1 | $312$ | $2$ | $2$ | $3$ |
| 312.48.3.fv.1 | $312$ | $2$ | $2$ | $3$ |
| 312.48.3.gb.1 | $312$ | $2$ | $2$ | $3$ |
| 312.48.3.gh.1 | $312$ | $2$ | $2$ | $3$ |