Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 4 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $8^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $4$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-11,-19,-27,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 8F0 |
| Sutherland and Zywina (SZ) label: | 8F0-8a |
| Rouse and Zureick-Brown (RZB) label: | X55 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.16.0.1 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}4&5\\1&4\end{bmatrix}$, $\begin{bmatrix}4&5\\3&7\end{bmatrix}$, $\begin{bmatrix}5&3\\5&2\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.D_6$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $12$ |
| Cyclic 8-torsion field degree: | $48$ |
| Full 8-torsion field degree: | $96$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^{18}\cdot3\,\frac{y(x+6y)^{19}(x^{2}-12xy-72y^{2})^{3}(x^{2}+12xy+72y^{2})^{3}}{(x+6y)^{16}(x^{2}+24xy+72y^{2})^{8}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{ns}}(8)$ | $8$ | $2$ | $2$ | $1$ |
| 8.32.1.b.1 | $8$ | $2$ | $2$ | $1$ |
| 8.32.1.c.1 | $8$ | $2$ | $2$ | $1$ |
| 8.32.1.d.1 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1.bs.1 | $8$ | $3$ | $3$ | $1$ |
| $X_{\mathrm{ns}}^+(16)$ | $16$ | $4$ | $4$ | $2$ |
| 24.32.1.e.1 | $24$ | $2$ | $2$ | $1$ |
| 24.32.1.f.1 | $24$ | $2$ | $2$ | $1$ |
| 24.32.1.g.1 | $24$ | $2$ | $2$ | $1$ |
| 24.32.1.h.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.mk.1 | $24$ | $3$ | $3$ | $1$ |
| 24.64.4.a.1 | $24$ | $4$ | $4$ | $4$ |
| 40.32.1.a.1 | $40$ | $2$ | $2$ | $1$ |
| 40.32.1.b.1 | $40$ | $2$ | $2$ | $1$ |
| 40.32.1.c.1 | $40$ | $2$ | $2$ | $1$ |
| 40.32.1.d.1 | $40$ | $2$ | $2$ | $1$ |
| 40.80.5.i.1 | $40$ | $5$ | $5$ | $5$ |
| 40.96.5.y.1 | $40$ | $6$ | $6$ | $5$ |
| 40.160.10.a.1 | $40$ | $10$ | $10$ | $10$ |
| 56.32.1.a.1 | $56$ | $2$ | $2$ | $1$ |
| 56.32.1.b.1 | $56$ | $2$ | $2$ | $1$ |
| 56.32.1.c.1 | $56$ | $2$ | $2$ | $1$ |
| 56.32.1.d.1 | $56$ | $2$ | $2$ | $1$ |
| 56.128.9.a.1 | $56$ | $8$ | $8$ | $9$ |
| 56.336.21.jq.1 | $56$ | $21$ | $21$ | $21$ |
| 56.448.30.a.1 | $56$ | $28$ | $28$ | $30$ |
| 88.32.1.a.1 | $88$ | $2$ | $2$ | $1$ |
| 88.32.1.b.1 | $88$ | $2$ | $2$ | $1$ |
| 88.32.1.c.1 | $88$ | $2$ | $2$ | $1$ |
| 88.32.1.d.1 | $88$ | $2$ | $2$ | $1$ |
| 88.192.15.a.1 | $88$ | $12$ | $12$ | $15$ |
| 104.32.1.a.1 | $104$ | $2$ | $2$ | $1$ |
| 104.32.1.b.1 | $104$ | $2$ | $2$ | $1$ |
| 104.32.1.c.1 | $104$ | $2$ | $2$ | $1$ |
| 104.32.1.d.1 | $104$ | $2$ | $2$ | $1$ |
| 104.224.15.a.1 | $104$ | $14$ | $14$ | $15$ |
| 120.32.1.e.1 | $120$ | $2$ | $2$ | $1$ |
| 120.32.1.f.1 | $120$ | $2$ | $2$ | $1$ |
| 120.32.1.g.1 | $120$ | $2$ | $2$ | $1$ |
| 120.32.1.h.1 | $120$ | $2$ | $2$ | $1$ |
| 136.32.1.a.1 | $136$ | $2$ | $2$ | $1$ |
| 136.32.1.b.1 | $136$ | $2$ | $2$ | $1$ |
| 136.32.1.c.1 | $136$ | $2$ | $2$ | $1$ |
| 136.32.1.d.1 | $136$ | $2$ | $2$ | $1$ |
| 136.288.21.bc.1 | $136$ | $18$ | $18$ | $21$ |
| 152.32.1.a.1 | $152$ | $2$ | $2$ | $1$ |
| 152.32.1.b.1 | $152$ | $2$ | $2$ | $1$ |
| 152.32.1.c.1 | $152$ | $2$ | $2$ | $1$ |
| 152.32.1.d.1 | $152$ | $2$ | $2$ | $1$ |
| 168.32.1.e.1 | $168$ | $2$ | $2$ | $1$ |
| 168.32.1.f.1 | $168$ | $2$ | $2$ | $1$ |
| 168.32.1.g.1 | $168$ | $2$ | $2$ | $1$ |
| 168.32.1.h.1 | $168$ | $2$ | $2$ | $1$ |
| 184.32.1.a.1 | $184$ | $2$ | $2$ | $1$ |
| 184.32.1.b.1 | $184$ | $2$ | $2$ | $1$ |
| 184.32.1.c.1 | $184$ | $2$ | $2$ | $1$ |
| 184.32.1.d.1 | $184$ | $2$ | $2$ | $1$ |
| 232.32.1.a.1 | $232$ | $2$ | $2$ | $1$ |
| 232.32.1.b.1 | $232$ | $2$ | $2$ | $1$ |
| 232.32.1.c.1 | $232$ | $2$ | $2$ | $1$ |
| 232.32.1.d.1 | $232$ | $2$ | $2$ | $1$ |
| 248.32.1.a.1 | $248$ | $2$ | $2$ | $1$ |
| 248.32.1.b.1 | $248$ | $2$ | $2$ | $1$ |
| 248.32.1.c.1 | $248$ | $2$ | $2$ | $1$ |
| 248.32.1.d.1 | $248$ | $2$ | $2$ | $1$ |
| 264.32.1.e.1 | $264$ | $2$ | $2$ | $1$ |
| 264.32.1.f.1 | $264$ | $2$ | $2$ | $1$ |
| 264.32.1.g.1 | $264$ | $2$ | $2$ | $1$ |
| 264.32.1.h.1 | $264$ | $2$ | $2$ | $1$ |
| 280.32.1.a.1 | $280$ | $2$ | $2$ | $1$ |
| 280.32.1.b.1 | $280$ | $2$ | $2$ | $1$ |
| 280.32.1.c.1 | $280$ | $2$ | $2$ | $1$ |
| 280.32.1.d.1 | $280$ | $2$ | $2$ | $1$ |
| 296.32.1.a.1 | $296$ | $2$ | $2$ | $1$ |
| 296.32.1.b.1 | $296$ | $2$ | $2$ | $1$ |
| 296.32.1.c.1 | $296$ | $2$ | $2$ | $1$ |
| 296.32.1.d.1 | $296$ | $2$ | $2$ | $1$ |
| 312.32.1.e.1 | $312$ | $2$ | $2$ | $1$ |
| 312.32.1.f.1 | $312$ | $2$ | $2$ | $1$ |
| 312.32.1.g.1 | $312$ | $2$ | $2$ | $1$ |
| 312.32.1.h.1 | $312$ | $2$ | $2$ | $1$ |
| 328.32.1.a.1 | $328$ | $2$ | $2$ | $1$ |
| 328.32.1.b.1 | $328$ | $2$ | $2$ | $1$ |
| 328.32.1.c.1 | $328$ | $2$ | $2$ | $1$ |
| 328.32.1.d.1 | $328$ | $2$ | $2$ | $1$ |