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: | X67c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.39 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&6\\6&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}7&0\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:C_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.g.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 41 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{x^{24}(x^{4}-4x^{3}y+4x^{2}y^{2}+3y^{4})^{3}(x^{4}-4x^{3}y+8x^{2}y^{2}-8xy^{3}+7y^{4})^{3}}{y^{8}x^{24}(x-y)^{8}(x^{2}+y^{2})^{2}(x^{2}-4xy+5y^{2})^{2}}$ |
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.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.b.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.d.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.2.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.o.1.1 | $8$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bm.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bn.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bq.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.br.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.cf.1.1 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.cj.1.2 | $24$ | $4$ | $4$ | $3$ |
| 40.96.1-40.bm.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bn.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bq.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.br.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.t.1.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.bp.1.1 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.cf.1.1 | $40$ | $10$ | $10$ | $15$ |
| 56.96.1-56.bm.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bn.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bq.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.br.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.bl.1.1 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.cf.1.1 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.ch.1.1 | $56$ | $28$ | $28$ | $45$ |
| 88.96.1-88.bm.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bn.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bq.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.br.1.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bm.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bn.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bq.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.br.1.1 | $104$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fi.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fj.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fm.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fn.1.1 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bm.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bn.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bq.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.br.1.1 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bm.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bn.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bq.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.br.1.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fi.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fj.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fm.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fn.1.1 | $168$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bm.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bn.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bq.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.br.1.1 | $184$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bm.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bn.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bq.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.br.1.1 | $232$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bm.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bn.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bq.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.br.1.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fi.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fj.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fm.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fn.1.1 | $264$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fe.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ff.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fi.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fj.1.1 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bm.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bn.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bq.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.br.1.1 | $296$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fi.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fj.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fm.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fn.1.1 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bm.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bn.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bq.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.br.1.1 | $328$ | $2$ | $2$ | $1$ |