Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.202 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&6\\0&1\end{bmatrix}$, $\begin{bmatrix}1&7\\4&7\end{bmatrix}$, $\begin{bmatrix}3&2\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_4^2:C_2$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.w.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $8$ |
| Full 8-torsion field degree: | $32$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ x^{2} + 16 y^{2} + z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.96.1-8.bk.1.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.bl.1.2 | $8$ | $2$ | $2$ | $1$ |
| 16.96.1-16.m.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.m.1.3 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.n.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.n.1.3 | $16$ | $2$ | $2$ | $1$ |
| 24.96.1-24.fk.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.fl.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.eu.1.9 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.eq.1.6 | $24$ | $4$ | $4$ | $3$ |
| 40.96.1-40.em.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.en.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.ce.1.5 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.dx.1.5 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.fc.1.3 | $40$ | $10$ | $10$ | $15$ |
| 48.96.1-48.m.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.m.1.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.n.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.n.1.6 | $48$ | $2$ | $2$ | $1$ |
| 56.96.1-56.em.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.en.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.dq.1.4 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.eu.1.4 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.ey.1.5 | $56$ | $28$ | $28$ | $45$ |
| 80.96.1-80.m.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.m.1.6 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.n.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.n.1.6 | $80$ | $2$ | $2$ | $1$ |
| 88.96.1-88.em.1.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.en.1.2 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1-104.em.1.4 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.en.1.4 | $104$ | $2$ | $2$ | $1$ |
| 112.96.1-112.m.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.m.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.n.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.n.1.5 | $112$ | $2$ | $2$ | $1$ |
| 120.96.1-120.oq.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.or.1.5 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1-136.em.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.en.1.3 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1-152.em.1.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.en.1.2 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1-168.oq.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.or.1.3 | $168$ | $2$ | $2$ | $1$ |
| 176.96.1-176.m.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.m.1.7 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.n.1.3 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.n.1.6 | $176$ | $2$ | $2$ | $1$ |
| 184.96.1-184.em.1.3 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.en.1.2 | $184$ | $2$ | $2$ | $1$ |
| 208.96.1-208.m.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.m.1.6 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.n.1.3 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.n.1.6 | $208$ | $2$ | $2$ | $1$ |
| 232.96.1-232.em.1.3 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.en.1.3 | $232$ | $2$ | $2$ | $1$ |
| 240.96.1-240.m.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.m.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.n.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.n.1.9 | $240$ | $2$ | $2$ | $1$ |
| 248.96.1-248.em.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.en.1.2 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1-264.oq.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.or.1.2 | $264$ | $2$ | $2$ | $1$ |
| 272.96.1-272.m.1.3 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.m.1.5 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.n.1.3 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.n.1.5 | $272$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ns.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.nt.1.2 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1-296.em.1.4 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.en.1.3 | $296$ | $2$ | $2$ | $1$ |
| 304.96.1-304.m.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.m.1.7 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.n.1.3 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.n.1.6 | $304$ | $2$ | $2$ | $1$ |
| 312.96.1-312.oq.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.or.1.3 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1-328.em.1.4 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.en.1.4 | $328$ | $2$ | $2$ | $1$ |