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 | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse and Zureick-Brown (RZB) label: | X66b |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.153 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&2\\4&1\end{bmatrix}$, $\begin{bmatrix}5&6\\2&3\end{bmatrix}$, $\begin{bmatrix}7&4\\4&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:C_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.a.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $16$ |
| Full 8-torsion field degree: | $32$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(x-2y)^{24}(x^{4}-8x^{3}y+64x^{2}y^{2}-192xy^{3}+192y^{4})^{3}(3x^{4}-24x^{3}y+64x^{2}y^{2}-64xy^{3}+64y^{4})^{3}}{(x-2y)^{24}(x^{2}-8y^{2})^{4}(x^{2}-8xy+8y^{2})^{4}(x^{2}-4xy+8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.a.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.a.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.c.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.d.1.2 | $8$ | $2$ | $2$ | $1$ |
| 16.96.0-16.a.1.4 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.b.1.4 | $16$ | $2$ | $2$ | $0$ |
| 16.96.2-16.a.1.4 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2-16.b.1.4 | $16$ | $2$ | $2$ | $2$ |
| 24.96.1-24.c.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.d.1.3 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.a.1.15 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.d.1.16 | $24$ | $4$ | $4$ | $3$ |
| 40.96.1-40.c.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.d.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.a.1.6 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.a.1.10 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.a.1.12 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.a.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.b.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.2-48.a.1.8 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.b.1.8 | $48$ | $2$ | $2$ | $2$ |
| 56.96.1-56.c.1.4 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.d.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.a.1.18 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.a.1.16 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.a.1.16 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.a.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.b.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.96.2-80.a.1.8 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-80.b.1.6 | $80$ | $2$ | $2$ | $2$ |
| 88.96.1-88.c.1.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.d.1.2 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1-104.c.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.d.1.4 | $104$ | $2$ | $2$ | $1$ |
| 112.96.0-112.a.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.b.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.2-112.a.1.6 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.b.1.8 | $112$ | $2$ | $2$ | $2$ |
| 120.96.1-120.c.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.d.1.4 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1-136.c.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.d.1.4 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1-152.c.1.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.d.1.2 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1-168.c.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.d.1.6 | $168$ | $2$ | $2$ | $1$ |
| 176.96.0-176.a.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.b.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.2-176.a.1.6 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.b.1.8 | $176$ | $2$ | $2$ | $2$ |
| 184.96.1-184.c.1.4 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.d.1.2 | $184$ | $2$ | $2$ | $1$ |
| 208.96.0-208.a.1.8 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.b.1.8 | $208$ | $2$ | $2$ | $0$ |
| 208.96.2-208.a.1.6 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2-208.b.1.8 | $208$ | $2$ | $2$ | $2$ |
| 232.96.1-232.c.1.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.d.1.4 | $232$ | $2$ | $2$ | $1$ |
| 240.96.0-240.a.1.16 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.b.1.16 | $240$ | $2$ | $2$ | $0$ |
| 240.96.2-240.a.1.16 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.b.1.14 | $240$ | $2$ | $2$ | $2$ |
| 248.96.1-248.c.1.4 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.d.1.2 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1-264.c.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.d.1.7 | $264$ | $2$ | $2$ | $1$ |
| 272.96.0-272.a.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.b.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.2-272.a.1.6 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.b.1.6 | $272$ | $2$ | $2$ | $2$ |
| 280.96.1-280.c.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.d.1.8 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1-296.c.1.2 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.d.1.4 | $296$ | $2$ | $2$ | $1$ |
| 304.96.0-304.a.1.8 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.b.1.8 | $304$ | $2$ | $2$ | $0$ |
| 304.96.2-304.a.1.6 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2-304.b.1.8 | $304$ | $2$ | $2$ | $2$ |
| 312.96.1-312.c.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.d.1.8 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1-328.c.1.2 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.d.1.4 | $328$ | $2$ | $2$ | $1$ |