Invariants
| Level: | $28$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-28$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.12.0.18 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}7&18\\10&11\end{bmatrix}$, $\begin{bmatrix}12&17\\25&0\end{bmatrix}$, $\begin{bmatrix}25&2\\18&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 28-isogeny field degree: | $16$ |
| Cyclic 28-torsion field degree: | $192$ |
| Full 28-torsion field degree: | $16128$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 64 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{x^{12}(112x^{2}+3y^{2})^{3}(112x^{2}+27y^{2})^{3}}{x^{12}(112x^{2}-9y^{2})^{4}(112x^{2}+9y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 28.6.0.a.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 28.6.0.c.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 28.24.0.h.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0.i.1 | $28$ | $2$ | $2$ | $0$ |
| 28.96.5.p.1 | $28$ | $8$ | $8$ | $5$ |
| 28.252.16.bv.1 | $28$ | $21$ | $21$ | $16$ |
| 28.336.21.bv.1 | $28$ | $28$ | $28$ | $21$ |
| 56.24.0.ca.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cb.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.co.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cp.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cu.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.cv.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.1.bc.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.be.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.do.1 | $56$ | $2$ | $2$ | $1$ |
| 56.24.1.dq.1 | $56$ | $2$ | $2$ | $1$ |
| 84.24.0.u.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0.v.1 | $84$ | $2$ | $2$ | $0$ |
| 84.36.2.fl.1 | $84$ | $3$ | $3$ | $2$ |
| 84.48.1.br.1 | $84$ | $4$ | $4$ | $1$ |
| 140.24.0.q.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0.r.1 | $140$ | $2$ | $2$ | $0$ |
| 140.60.4.br.1 | $140$ | $5$ | $5$ | $4$ |
| 140.72.3.ed.1 | $140$ | $6$ | $6$ | $3$ |
| 140.120.7.gp.1 | $140$ | $10$ | $10$ | $7$ |
| 168.24.0.fs.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ft.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.gg.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.gh.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.gm.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.gn.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.1.dw.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.dy.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mp.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mr.1 | $168$ | $2$ | $2$ | $1$ |
| 252.324.22.fp.1 | $252$ | $27$ | $27$ | $22$ |
| 280.24.0.fq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ge.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gk.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gl.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.dw.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.dy.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mm.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mo.1 | $280$ | $2$ | $2$ | $1$ |
| 308.24.0.k.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0.l.1 | $308$ | $2$ | $2$ | $0$ |
| 308.144.9.bj.1 | $308$ | $12$ | $12$ | $9$ |