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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.12.0.11 |
Level structure
$\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}12&1\\27&20\end{bmatrix}$, $\begin{bmatrix}19&2\\4&13\end{bmatrix}$ |
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
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + 3 x z + 112 y^{2} + 2 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
4.6.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
14.6.0.b.1 | $14$ | $2$ | $2$ | $0$ | $0$ |
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 |
---|---|---|---|---|
28.96.5.m.1 | $28$ | $8$ | $8$ | $5$ |
28.252.16.bs.1 | $28$ | $21$ | $21$ | $16$ |
28.336.21.bs.1 | $28$ | $28$ | $28$ | $21$ |
56.24.1.q.1 | $56$ | $2$ | $2$ | $1$ |
56.24.1.s.1 | $56$ | $2$ | $2$ | $1$ |
56.24.1.dc.1 | $56$ | $2$ | $2$ | $1$ |
56.24.1.de.1 | $56$ | $2$ | $2$ | $1$ |
84.36.2.fi.1 | $84$ | $3$ | $3$ | $2$ |
84.48.1.bo.1 | $84$ | $4$ | $4$ | $1$ |
140.60.4.bo.1 | $140$ | $5$ | $5$ | $4$ |
140.72.3.ea.1 | $140$ | $6$ | $6$ | $3$ |
140.120.7.gm.1 | $140$ | $10$ | $10$ | $7$ |
168.24.1.dk.1 | $168$ | $2$ | $2$ | $1$ |
168.24.1.dm.1 | $168$ | $2$ | $2$ | $1$ |
168.24.1.md.1 | $168$ | $2$ | $2$ | $1$ |
168.24.1.mf.1 | $168$ | $2$ | $2$ | $1$ |
252.324.22.fm.1 | $252$ | $27$ | $27$ | $22$ |
280.24.1.dk.1 | $280$ | $2$ | $2$ | $1$ |
280.24.1.dm.1 | $280$ | $2$ | $2$ | $1$ |
280.24.1.ma.1 | $280$ | $2$ | $2$ | $1$ |
280.24.1.mc.1 | $280$ | $2$ | $2$ | $1$ |
308.144.9.bg.1 | $308$ | $12$ | $12$ | $9$ |