Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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 $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
| Rouse and Zureick-Brown (RZB) label: | X32i |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.57 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\0&5\end{bmatrix}$, $\begin{bmatrix}3&1\\4&3\end{bmatrix}$, $\begin{bmatrix}5&7\\4&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.D_4$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.o.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $64$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1491 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}+16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x^{2}+16y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0-4.c.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.j.1.4 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.w.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.y.1.4 | $8$ | $2$ | $2$ | $0$ |
| 16.48.0-16.i.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.48.0-16.j.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.48.1-16.c.1.6 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1-16.d.1.6 | $16$ | $2$ | $2$ | $1$ |
| 24.48.0-24.bo.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bq.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bs.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bu.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-24.cw.1.5 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-24.iw.1.18 | $24$ | $4$ | $4$ | $1$ |
| 40.48.0-40.bs.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bu.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bw.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.by.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-40.bq.1.11 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-40.cg.1.22 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-40.cw.1.11 | $40$ | $10$ | $10$ | $7$ |
| 48.48.0-48.i.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.j.1.16 | $48$ | $2$ | $2$ | $0$ |
| 48.48.1-48.c.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-48.d.1.9 | $48$ | $2$ | $2$ | $1$ |
| 56.48.0-56.bm.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bo.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bq.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bs.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.192.5-56.bq.1.3 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.cw.1.6 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.cw.1.19 | $56$ | $28$ | $28$ | $21$ |
| 80.48.0-80.q.1.10 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.r.1.12 | $80$ | $2$ | $2$ | $0$ |
| 80.48.1-80.c.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-80.d.1.7 | $80$ | $2$ | $2$ | $1$ |
| 88.48.0-88.bm.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bo.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bq.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bs.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-88.bq.1.10 | $88$ | $12$ | $12$ | $9$ |
| 104.48.0-104.bs.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bu.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bw.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.by.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.cg.1.7 | $104$ | $14$ | $14$ | $11$ |
| 112.48.0-112.i.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.j.1.8 | $112$ | $2$ | $2$ | $0$ |
| 112.48.1-112.c.1.9 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.d.1.11 | $112$ | $2$ | $2$ | $1$ |
| 120.48.0-120.dw.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dy.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ee.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eg.1.13 | $120$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bs.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bu.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bw.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.by.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.432.15-136.cs.1.7 | $136$ | $18$ | $18$ | $15$ |
| 152.48.0-152.bm.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bo.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bq.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bs.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.480.17-152.bq.1.6 | $152$ | $20$ | $20$ | $17$ |
| 168.48.0-168.dq.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ds.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dy.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ea.1.2 | $168$ | $2$ | $2$ | $0$ |
| 176.48.0-176.i.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.48.0-176.j.1.8 | $176$ | $2$ | $2$ | $0$ |
| 176.48.1-176.c.1.9 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1-176.d.1.13 | $176$ | $2$ | $2$ | $1$ |
| 184.48.0-184.bm.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bo.1.7 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bq.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bs.1.7 | $184$ | $2$ | $2$ | $0$ |
| 208.48.0-208.q.1.4 | $208$ | $2$ | $2$ | $0$ |
| 208.48.0-208.r.1.8 | $208$ | $2$ | $2$ | $0$ |
| 208.48.1-208.c.1.9 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1-208.d.1.13 | $208$ | $2$ | $2$ | $1$ |
| 232.48.0-232.bs.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bu.1.7 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bw.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.by.1.7 | $232$ | $2$ | $2$ | $0$ |
| 240.48.0-240.q.1.18 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.r.1.20 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-240.c.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.d.1.13 | $240$ | $2$ | $2$ | $1$ |
| 248.48.0-248.bm.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bo.1.7 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bq.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bs.1.7 | $248$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dq.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ds.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dy.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ea.1.15 | $264$ | $2$ | $2$ | $0$ |
| 272.48.0-272.q.1.8 | $272$ | $2$ | $2$ | $0$ |
| 272.48.0-272.r.1.4 | $272$ | $2$ | $2$ | $0$ |
| 272.48.1-272.c.1.13 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1-272.d.1.9 | $272$ | $2$ | $2$ | $1$ |
| 280.48.0-280.dw.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dy.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ee.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.eg.1.13 | $280$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bs.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bu.1.7 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bw.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.by.1.7 | $296$ | $2$ | $2$ | $0$ |
| 304.48.0-304.i.1.4 | $304$ | $2$ | $2$ | $0$ |
| 304.48.0-304.j.1.8 | $304$ | $2$ | $2$ | $0$ |
| 304.48.1-304.c.1.9 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1-304.d.1.13 | $304$ | $2$ | $2$ | $1$ |
| 312.48.0-312.dw.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dy.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ee.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.eg.1.2 | $312$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bs.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bu.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bw.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.by.1.1 | $328$ | $2$ | $2$ | $0$ |