Invariants
| Level: | $12$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.0.14 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&8\\10&1\end{bmatrix}$, $\begin{bmatrix}5&2\\4&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.c.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $96$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 54 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{(3x+y)^{24}(144x^{4}+144x^{3}y+72x^{2}y^{2}+12xy^{3}+y^{4})^{3}(1872x^{4}+2160x^{3}y+936x^{2}y^{2}+180xy^{3}+13y^{4})^{3}}{(2x+y)^{4}(3x+y)^{24}(6x+y)^{4}(12x^{2}-y^{2})^{4}(12x^{2}+6xy+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.b.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-4.b.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.a.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.b.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 12.24.0-12.b.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 12.144.4-12.e.1.5 | $12$ | $3$ | $3$ | $4$ |
| 12.192.3-12.e.1.1 | $12$ | $4$ | $4$ | $3$ |
| 24.96.0-24.g.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.g.2.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.h.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.h.2.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.i.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.i.2.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.j.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.j.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.p.1.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.u.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bs.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bu.1.1 | $24$ | $2$ | $2$ | $1$ |
| 36.1296.46-36.g.1.4 | $36$ | $27$ | $27$ | $46$ |
| 60.240.8-60.c.1.7 | $60$ | $5$ | $5$ | $8$ |
| 60.288.7-60.r.1.2 | $60$ | $6$ | $6$ | $7$ |
| 60.480.15-60.c.1.9 | $60$ | $10$ | $10$ | $15$ |
| 84.384.11-84.c.1.1 | $84$ | $8$ | $8$ | $11$ |
| 120.96.0-120.g.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.g.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.h.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.h.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.i.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.i.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.j.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.j.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.bs.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bu.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.cy.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.da.1.1 | $120$ | $2$ | $2$ | $1$ |
| 168.96.0-168.g.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.g.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.h.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.h.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.i.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.i.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.j.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.j.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.bs.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bu.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.cy.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.da.1.2 | $168$ | $2$ | $2$ | $1$ |
| 264.96.0-264.g.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.g.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.h.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.h.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.i.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.i.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.j.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.j.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.bs.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bu.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.cy.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.da.1.2 | $264$ | $2$ | $2$ | $1$ |
| 312.96.0-312.g.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.g.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.h.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.h.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.i.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.i.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.j.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.j.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.bs.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bu.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.cy.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.da.1.1 | $312$ | $2$ | $2$ | $1$ |