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 | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse and Zureick-Brown (RZB) label: | X61h |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.40 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\4&5\end{bmatrix}$, $\begin{bmatrix}3&2\\2&1\end{bmatrix}$, $\begin{bmatrix}5&6\\4&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:C_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.f.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $8$ |
| 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 41 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{x^{24}(x^{2}-xy+y^{2})^{3}(x^{2}+xy+y^{2})^{3}(x^{4}-x^{2}y^{2}+y^{4})^{3}}{y^{8}x^{32}(x^{4}+y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.a.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.a.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.a.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.c.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.f.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.g.2.2 | $8$ | $2$ | $2$ | $1$ |
| 16.96.0-16.c.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.c.2.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.2-16.c.1.2 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2-16.c.2.2 | $16$ | $2$ | $2$ | $2$ |
| 24.96.1-24.bk.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bl.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bo.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bp.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.ce.1.1 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.ci.1.4 | $24$ | $4$ | $4$ | $3$ |
| 40.96.1-40.bk.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bl.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bo.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bp.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.s.1.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.bo.1.1 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.ce.1.1 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.c.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.c.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.2-48.c.1.2 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.c.2.2 | $48$ | $2$ | $2$ | $2$ |
| 56.96.1-56.bk.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bl.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bo.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bp.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.bk.1.2 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.ce.1.1 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.cg.1.1 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.e.1.2 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.e.2.2 | $80$ | $2$ | $2$ | $0$ |
| 80.96.2-80.e.1.2 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-80.e.2.2 | $80$ | $2$ | $2$ | $2$ |
| 88.96.1-88.bk.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bl.1.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bo.1.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bp.1.2 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bk.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bl.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bo.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bp.1.2 | $104$ | $2$ | $2$ | $1$ |
| 112.96.0-112.c.1.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.c.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.2-112.c.1.2 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.c.2.2 | $112$ | $2$ | $2$ | $2$ |
| 120.96.1-120.fg.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fh.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fk.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fl.1.1 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bk.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bl.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bo.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bp.1.2 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bk.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bl.1.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bo.1.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bp.1.2 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fh.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fk.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fl.1.2 | $168$ | $2$ | $2$ | $1$ |
| 176.96.0-176.c.1.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.c.2.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.2-176.c.1.2 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.c.2.2 | $176$ | $2$ | $2$ | $2$ |
| 184.96.1-184.bk.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bl.1.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bo.1.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bp.1.2 | $184$ | $2$ | $2$ | $1$ |
| 208.96.0-208.e.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.e.2.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.2-208.e.1.2 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2-208.e.2.2 | $208$ | $2$ | $2$ | $2$ |
| 232.96.1-232.bk.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bl.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bo.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bp.1.2 | $232$ | $2$ | $2$ | $1$ |
| 240.96.0-240.e.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.e.2.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.2-240.e.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.e.2.2 | $240$ | $2$ | $2$ | $2$ |
| 248.96.1-248.bk.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bl.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bo.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bp.1.2 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fg.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fh.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fk.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fl.1.2 | $264$ | $2$ | $2$ | $1$ |
| 272.96.0-272.e.1.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.e.2.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.2-272.e.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.e.2.1 | $272$ | $2$ | $2$ | $2$ |
| 280.96.1-280.fc.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fd.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fg.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fh.1.1 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bk.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bl.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bo.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bp.1.2 | $296$ | $2$ | $2$ | $1$ |
| 304.96.0-304.c.1.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.c.2.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.2-304.c.1.2 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2-304.c.2.2 | $304$ | $2$ | $2$ | $2$ |
| 312.96.1-312.fg.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fh.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fk.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fl.1.2 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bk.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bl.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bo.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bp.1.2 | $328$ | $2$ | $2$ | $1$ |