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: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse and Zureick-Brown (RZB) label: | X79i |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.185 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&5\\4&3\end{bmatrix}$, $\begin{bmatrix}7&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_4:C_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.y.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| 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 15 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\,\frac{(x-2y)^{24}(3x^{4}-48x^{3}y+272x^{2}y^{2}-640xy^{3}+704y^{4})^{3}(11x^{4}-80x^{3}y+272x^{2}y^{2}-384xy^{3}+192y^{4})^{3}}{(x-2y)^{24}(x^{2}-8y^{2})^{8}(x^{2}-8xy+8y^{2})^{2}(x^{2}-4xy+8y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.k.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.p.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.p.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 |
|---|---|---|---|---|
| 16.96.0-16.r.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.s.1.4 | $16$ | $2$ | $2$ | $0$ |
| 16.96.1-16.o.1.1 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.p.1.1 | $16$ | $2$ | $2$ | $1$ |
| 16.96.2-16.i.1.3 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2-16.o.1.5 | $16$ | $2$ | $2$ | $2$ |
| 24.144.4-24.ew.1.13 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.es.1.7 | $24$ | $4$ | $4$ | $3$ |
| 40.240.8-40.cg.1.4 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.dz.1.15 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.fe.1.12 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.r.1.6 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.s.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.o.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.p.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.2-48.j.1.12 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.o.1.12 | $48$ | $2$ | $2$ | $2$ |
| 56.384.11-56.ds.1.12 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.ew.1.6 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.fa.1.7 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.x.1.6 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.y.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.o.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.p.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.96.2-80.o.1.7 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-80.u.1.11 | $80$ | $2$ | $2$ | $2$ |
| 112.96.0-112.r.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.s.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.o.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.p.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.2-112.j.1.11 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.o.1.11 | $112$ | $2$ | $2$ | $2$ |
| 176.96.0-176.r.1.6 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.s.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.o.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.p.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.2-176.j.1.12 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.o.1.12 | $176$ | $2$ | $2$ | $2$ |
| 208.96.0-208.x.1.6 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.y.1.8 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-208.o.1.2 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.p.1.2 | $208$ | $2$ | $2$ | $1$ |
| 208.96.2-208.o.1.11 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2-208.u.1.11 | $208$ | $2$ | $2$ | $2$ |
| 240.96.0-240.x.1.12 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.y.1.16 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.o.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.p.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-240.o.1.15 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.u.1.23 | $240$ | $2$ | $2$ | $2$ |
| 272.96.0-272.x.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.y.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.o.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.p.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.96.2-272.n.1.12 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.u.1.12 | $272$ | $2$ | $2$ | $2$ |
| 304.96.0-304.r.1.6 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.s.1.8 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.o.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.p.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.2-304.j.1.12 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2-304.o.1.12 | $304$ | $2$ | $2$ | $2$ |