Invariants
| Level: | $28$ | $\SL_2$-level: | $28$ | Newform level: | $784$ | ||
| Index: | $168$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $8 = 1 + \frac{ 168 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $28^{6}$ | Cusp orbits | $6$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-11,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 28B8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.168.8.2 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}7&24\\12&11\end{bmatrix}$, $\begin{bmatrix}9&10\\21&19\end{bmatrix}$, $\begin{bmatrix}19&13\\20&9\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 28-isogeny field degree: | $48$ |
| Cyclic 28-torsion field degree: | $576$ |
| Full 28-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{30}\cdot7^{16}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{6}\cdot2$ |
| Newforms: | 196.2.a.a, 784.2.a.a, 784.2.a.b, 784.2.a.f, 784.2.a.g, 784.2.a.h, 784.2.a.k |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x^{2} - x y - 2 x z - x w + x u + y w - y u + 2 z w - z u - z v - w u + w v $ |
| $=$ | $x y + 2 x z - x w - x t - x r - 3 y z + y w + y t - 2 z^{2} - z u - z v + z r - 2 t v - u v - v^{2}$ | |
| $=$ | $x y - x w - 2 x r - y^{2} - 2 y z + y w + y u + y v + 2 z w + 2 w v - 2 t u - 2 t v - u r - v r$ | |
| $=$ | $x^{2} + x w + 2 x u + 2 x v - x r - y^{2} - y z + 2 y u + y v + z w - w u - w v - 2 t u + u^{2} + \cdots + v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 162957 x^{14} - 2592048 x^{13} y - 2024800 x^{13} z + 16575774 x^{12} y^{2} + 30661734 x^{12} y z + \cdots + 192 y^{2} z^{12} $ |
Rational points
This modular curve has 4 rational CM points but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 28.84.4.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -u-v$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}+12XY+5Y^{2}+12XZ+2YZ-Z^{2}-3XW-5YW-W^{2} $ |
| $=$ | $ X^{2}Y+5XY^{2}+2Y^{3}-3X^{2}Z-Y^{2}Z-3XZ^{2}-YZ^{2}+X^{2}W+XYW-Y^{2}W+5XZW+3YZW-Z^{2}W-XW^{2}-YW^{2} $ |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(14)$ | $14$ | $4$ | $4$ | $1$ | $1$ | $1^{5}\cdot2$ |
| 28.84.4.a.1 | $28$ | $2$ | $2$ | $4$ | $4$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(28)$ | $28$ | $2$ | $2$ | $23$ | $8$ | $1^{7}\cdot2^{4}$ |
| 28.336.23.d.1 | $28$ | $2$ | $2$ | $23$ | $11$ | $1^{7}\cdot2^{4}$ |
| 28.336.23.n.1 | $28$ | $2$ | $2$ | $23$ | $13$ | $1^{7}\cdot2^{4}$ |
| 28.336.23.p.1 | $28$ | $2$ | $2$ | $23$ | $12$ | $1^{7}\cdot2^{4}$ |
| 28.504.30.o.1 | $28$ | $3$ | $3$ | $30$ | $15$ | $1^{12}\cdot2^{5}$ |
| 56.336.23.c.1 | $56$ | $2$ | $2$ | $23$ | $14$ | $1^{7}\cdot2^{4}$ |
| 56.336.23.l.1 | $56$ | $2$ | $2$ | $23$ | $17$ | $1^{7}\cdot2^{4}$ |
| 56.336.23.bp.1 | $56$ | $2$ | $2$ | $23$ | $17$ | $1^{7}\cdot2^{4}$ |
| 56.336.23.bv.1 | $56$ | $2$ | $2$ | $23$ | $14$ | $1^{7}\cdot2^{4}$ |
| $X_{\mathrm{ns}}^+(56)$ | $56$ | $4$ | $4$ | $43$ | $43$ | $1^{15}\cdot2^{10}$ |
| 84.336.23.bz.1 | $84$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 84.336.23.cb.1 | $84$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 84.336.23.dz.1 | $84$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 84.336.23.eb.1 | $84$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 140.336.23.v.1 | $140$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 140.336.23.x.1 | $140$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 140.336.23.bt.1 | $140$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 140.336.23.bv.1 | $140$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 168.336.23.gr.1 | $168$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 168.336.23.gx.1 | $168$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 168.336.23.nl.1 | $168$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 168.336.23.nr.1 | $168$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 280.336.23.cn.1 | $280$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 280.336.23.ct.1 | $280$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 280.336.23.fh.1 | $280$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 280.336.23.fn.1 | $280$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 308.336.23.br.1 | $308$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 308.336.23.bt.1 | $308$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 308.336.23.bv.1 | $308$ | $2$ | $2$ | $23$ | $?$ | not computed |
| 308.336.23.bx.1 | $308$ | $2$ | $2$ | $23$ | $?$ | not computed |