Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse and Zureick-Brown (RZB) label: | X13d |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.12.0.6 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&1\\4&7\end{bmatrix}$, $\begin{bmatrix}7&1\\4&1\end{bmatrix}$, $\begin{bmatrix}7&2\\4&5\end{bmatrix}$, $\begin{bmatrix}7&5\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^4.D_4$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $128$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.24.0-4.b.1.5 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-4.d.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.d.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.k.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.m.1.5 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.m.1.7 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.n.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.n.1.6 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.o.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.o.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.p.1.5 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.p.1.7 | $8$ | $2$ | $2$ | $0$ |
| 24.24.0-12.g.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-12.h.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.36.1-12.c.1.4 | $24$ | $3$ | $3$ | $1$ |
| 24.48.0-12.g.1.19 | $24$ | $4$ | $4$ | $0$ |
| 40.24.0-20.g.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-20.h.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.60.2-20.c.1.10 | $40$ | $5$ | $5$ | $2$ |
| 40.72.1-20.c.1.21 | $40$ | $6$ | $6$ | $1$ |
| 40.120.3-20.c.1.1 | $40$ | $10$ | $10$ | $3$ |
| 24.24.0-24.s.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.v.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.y.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.y.1.14 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.z.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.z.1.12 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.ba.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.ba.1.12 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.bb.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.bb.1.14 | $24$ | $2$ | $2$ | $0$ |
| 56.24.0-28.g.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-28.h.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.2-28.c.1.25 | $56$ | $8$ | $8$ | $2$ |
| 56.252.7-28.c.1.21 | $56$ | $21$ | $21$ | $7$ |
| 56.336.9-28.c.1.9 | $56$ | $28$ | $28$ | $9$ |
| 72.324.10-36.d.1.11 | $72$ | $27$ | $27$ | $10$ |
| 40.24.0-40.s.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.v.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.y.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.y.1.13 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.z.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.z.1.10 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.ba.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.ba.1.10 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.bb.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.bb.1.13 | $40$ | $2$ | $2$ | $0$ |
| 88.24.0-44.g.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-44.h.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.144.4-44.c.1.24 | $88$ | $12$ | $12$ | $4$ |
| 104.24.0-52.g.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-52.h.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.168.5-52.c.1.17 | $104$ | $14$ | $14$ | $5$ |
| 56.24.0-56.s.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.v.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.y.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.y.1.13 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.z.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.z.1.10 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.ba.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.ba.1.10 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.bb.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.bb.1.13 | $56$ | $2$ | $2$ | $0$ |
| 120.24.0-60.g.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-60.h.1.11 | $120$ | $2$ | $2$ | $0$ |
| 136.24.0-68.g.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-68.h.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.216.7-68.c.1.22 | $136$ | $18$ | $18$ | $7$ |
| 152.24.0-76.g.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-76.h.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.240.8-76.c.1.26 | $152$ | $20$ | $20$ | $8$ |
| 168.24.0-84.g.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-84.h.1.10 | $168$ | $2$ | $2$ | $0$ |
| 88.24.0-88.s.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.v.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.y.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.y.1.11 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.z.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.z.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.ba.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.ba.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.bb.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.bb.1.11 | $88$ | $2$ | $2$ | $0$ |
| 184.24.0-92.g.1.6 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-92.h.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.288.10-92.c.1.23 | $184$ | $24$ | $24$ | $10$ |
| 104.24.0-104.s.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.v.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.y.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.y.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.z.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.z.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.ba.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.ba.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.bb.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.bb.1.7 | $104$ | $2$ | $2$ | $0$ |
| 232.24.0-116.g.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-116.h.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.360.13-116.c.1.17 | $232$ | $30$ | $30$ | $13$ |
| 120.24.0-120.s.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.v.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.y.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.y.1.25 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.z.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.z.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.ba.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.ba.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.bb.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.bb.1.25 | $120$ | $2$ | $2$ | $0$ |
| 248.24.0-124.g.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-124.h.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.384.14-124.c.1.22 | $248$ | $32$ | $32$ | $14$ |
| 264.24.0-132.g.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-132.h.1.10 | $264$ | $2$ | $2$ | $0$ |
| 136.24.0-136.s.1.7 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.v.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.y.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.y.1.16 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.z.1.14 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.z.1.16 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.ba.1.14 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.ba.1.16 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.bb.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.bb.1.16 | $136$ | $2$ | $2$ | $0$ |
| 280.24.0-140.g.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-140.h.1.12 | $280$ | $2$ | $2$ | $0$ |
| 296.24.0-148.g.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-148.h.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.456.17-148.c.1.23 | $296$ | $38$ | $38$ | $17$ |
| 152.24.0-152.s.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.v.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.y.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.y.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.z.1.2 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.z.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.ba.1.2 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.ba.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.bb.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.bb.1.7 | $152$ | $2$ | $2$ | $0$ |
| 312.24.0-156.g.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-156.h.1.10 | $312$ | $2$ | $2$ | $0$ |
| 328.24.0-164.g.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-164.h.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.504.19-164.c.1.24 | $328$ | $42$ | $42$ | $19$ |
| 168.24.0-168.s.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.v.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.y.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.y.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.z.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.z.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.ba.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.ba.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.bb.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.bb.1.21 | $168$ | $2$ | $2$ | $0$ |
| 184.24.0-184.s.1.7 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.v.1.7 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.y.1.5 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.y.1.13 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.z.1.2 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.z.1.10 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.ba.1.2 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.ba.1.10 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.bb.1.5 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.bb.1.13 | $184$ | $2$ | $2$ | $0$ |
| 232.24.0-232.s.1.7 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.v.1.7 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.y.1.9 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.y.1.13 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.z.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.z.1.11 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.ba.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.ba.1.11 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.bb.1.9 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.bb.1.13 | $232$ | $2$ | $2$ | $0$ |
| 248.24.0-248.s.1.7 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.v.1.7 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.y.1.3 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.y.1.7 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.z.1.2 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.z.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.ba.1.2 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.ba.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.bb.1.3 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.bb.1.7 | $248$ | $2$ | $2$ | $0$ |
| 264.24.0-264.s.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.v.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.y.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.y.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.z.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.z.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.ba.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.ba.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.bb.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.bb.1.13 | $264$ | $2$ | $2$ | $0$ |
| 280.24.0-280.s.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.v.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.y.1.17 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.y.1.21 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.z.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.z.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.ba.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.ba.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.bb.1.17 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.bb.1.21 | $280$ | $2$ | $2$ | $0$ |
| 296.24.0-296.s.1.7 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.v.1.7 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.y.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.y.1.7 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.z.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.z.1.6 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.ba.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.ba.1.6 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.bb.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.bb.1.7 | $296$ | $2$ | $2$ | $0$ |
| 312.24.0-312.s.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.v.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.y.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.y.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.z.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.z.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.ba.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.ba.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.bb.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.bb.1.21 | $312$ | $2$ | $2$ | $0$ |
| 328.24.0-328.s.1.7 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.v.1.8 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.y.1.8 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.y.1.16 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.z.1.12 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.z.1.16 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.ba.1.12 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.ba.1.16 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.bb.1.8 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.bb.1.16 | $328$ | $2$ | $2$ | $0$ |