Invariants
| Level: | $28$ | $\SL_2$-level: | $28$ | Newform level: | $784$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $28^{18}$ | Cusp orbits | $6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.1008.34.15 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}11&22\\6&3\end{bmatrix}$, $\begin{bmatrix}19&10\\18&9\end{bmatrix}$, $\begin{bmatrix}21&4\\12&21\end{bmatrix}$ |
| $\GL_2(\Z/28\Z)$-subgroup: | $C_6\times \SD_{32}$ |
| Contains $-I$: | no $\quad$ (see 28.504.34.d.1 for the level structure with $-I$) |
| Cyclic 28-isogeny field degree: | $16$ |
| Cyclic 28-torsion field degree: | $96$ |
| Full 28-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{86}\cdot7^{68}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}$ |
| Newforms: | 98.2.a.b$^{4}$, 196.2.a.a$^{2}$, 196.2.a.b, 196.2.a.c$^{3}$, 392.2.a.a$^{2}$, 392.2.a.e$^{2}$, 392.2.a.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=11,31,37,67,149$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.48.0-28.b.1.1 | $28$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 28.504.16-28.a.1.1 | $28$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
| 28.504.16-28.a.1.7 | $28$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
| 28.504.16-28.d.1.1 | $28$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
| 28.504.16-28.d.1.2 | $28$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
| 28.504.16-28.d.1.7 | $28$ | $2$ | $2$ | $16$ | $2$ | $1^{6}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.2016.67-28.k.1.1 | $28$ | $2$ | $2$ | $67$ | $14$ | $1^{27}\cdot2^{3}$ |
| 28.2016.67-28.m.1.1 | $28$ | $2$ | $2$ | $67$ | $25$ | $1^{27}\cdot2^{3}$ |
| 28.2016.67-28.q.1.1 | $28$ | $2$ | $2$ | $67$ | $18$ | $1^{27}\cdot2^{3}$ |
| 28.2016.67-28.s.1.2 | $28$ | $2$ | $2$ | $67$ | $18$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.bq.1.2 | $56$ | $2$ | $2$ | $67$ | $22$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.cc.1.2 | $56$ | $2$ | $2$ | $67$ | $25$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.cy.1.2 | $56$ | $2$ | $2$ | $67$ | $33$ | $1^{27}\cdot2^{3}$ |
| 56.2016.67-56.dk.1.2 | $56$ | $2$ | $2$ | $67$ | $13$ | $1^{27}\cdot2^{3}$ |
| 56.2016.73-56.cj.1.1 | $56$ | $2$ | $2$ | $73$ | $21$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.ck.1.1 | $56$ | $2$ | $2$ | $73$ | $25$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.ga.1.1 | $56$ | $2$ | $2$ | $73$ | $21$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.gb.1.2 | $56$ | $2$ | $2$ | $73$ | $28$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.ie.1.1 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.if.1.1 | $56$ | $2$ | $2$ | $73$ | $22$ | $1^{23}\cdot2^{8}$ |
| 56.2016.73-56.iz.1.2 | $56$ | $2$ | $2$ | $73$ | $26$ | $1^{7}\cdot2^{14}\cdot4$ |
| 56.2016.73-56.ja.1.1 | $56$ | $2$ | $2$ | $73$ | $22$ | $1^{7}\cdot2^{14}\cdot4$ |