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: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse and Zureick-Brown (RZB) label: | X75g |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.163 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&1\\0&7\end{bmatrix}$, $\begin{bmatrix}3&7\\0&5\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_8$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.k.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| 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 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2}\cdot\frac{x^{24}(x^{4}-16x^{3}y-112x^{2}y^{2}-128xy^{3}+64y^{4})^{3}(x^{4}+16x^{3}y-112x^{2}y^{2}+128xy^{3}+64y^{4})^{3}}{y^{2}x^{26}(x^{2}-8y^{2})^{2}(x^{2}+8y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.d.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.1.12 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.p.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.p.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.96.0-8.m.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.m.2.3 | $8$ | $2$ | $2$ | $0$ |
| 16.96.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.f.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.1-16.d.1.5 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.f.1.5 | $16$ | $2$ | $2$ | $1$ |
| 24.96.0-24.bd.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bd.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.144.4-24.cp.1.1 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.cr.1.2 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.be.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.be.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.240.8-40.z.1.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.bx.1.1 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.cp.1.3 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.e.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.f.1.3 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.d.1.10 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.f.1.10 | $48$ | $2$ | $2$ | $1$ |
| 56.96.0-56.bc.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bc.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.384.11-56.bt.1.3 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.cp.1.1 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.cr.1.2 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.g.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.h.1.2 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.d.1.9 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.f.1.5 | $80$ | $2$ | $2$ | $1$ |
| 88.96.0-88.bc.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bc.2.3 | $88$ | $2$ | $2$ | $0$ |
| 104.96.0-104.be.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.be.2.3 | $104$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.d.1.6 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.f.1.6 | $112$ | $2$ | $2$ | $1$ |
| 120.96.0-120.db.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.db.2.5 | $120$ | $2$ | $2$ | $0$ |
| 136.96.0-136.be.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.be.2.3 | $136$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bc.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bc.2.3 | $152$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.2.5 | $168$ | $2$ | $2$ | $0$ |
| 176.96.0-176.e.1.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.f.1.3 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.d.1.6 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.f.1.6 | $176$ | $2$ | $2$ | $1$ |
| 184.96.0-184.bc.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bc.2.3 | $184$ | $2$ | $2$ | $0$ |
| 208.96.0-208.g.1.3 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.h.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-208.d.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.f.1.5 | $208$ | $2$ | $2$ | $1$ |
| 232.96.0-232.be.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.be.2.5 | $232$ | $2$ | $2$ | $0$ |
| 240.96.0-240.g.1.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.h.1.5 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.d.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.f.1.18 | $240$ | $2$ | $2$ | $1$ |
| 248.96.0-248.bc.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bc.2.5 | $248$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cz.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cz.2.3 | $264$ | $2$ | $2$ | $0$ |
| 272.96.0-272.g.1.4 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.h.1.4 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.d.1.11 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.f.1.7 | $272$ | $2$ | $2$ | $1$ |
| 280.96.0-280.da.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.da.2.5 | $280$ | $2$ | $2$ | $0$ |
| 296.96.0-296.be.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.be.2.3 | $296$ | $2$ | $2$ | $0$ |
| 304.96.0-304.e.1.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.f.1.3 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.d.1.6 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.f.1.6 | $304$ | $2$ | $2$ | $1$ |
| 312.96.0-312.db.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.db.2.5 | $312$ | $2$ | $2$ | $0$ |
| 328.96.0-328.be.1.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.be.2.3 | $328$ | $2$ | $2$ | $0$ |