Invariants
| Level: | $8$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse and Zureick-Brown (RZB) label: | X25g |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.20 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}1&6\\4&1\end{bmatrix}$, $\begin{bmatrix}3&6\\4&5\end{bmatrix}$, $\begin{bmatrix}7&0\\4&3\end{bmatrix}$, $\begin{bmatrix}7&2\\4&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_4\times C_2^3$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.b.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $8$ |
| 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 2569 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(256x^{4}-16x^{2}y^{2}+y^{4})^{3}}{y^{4}x^{16}(4x-y)^{2}(4x+y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.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 |
|---|---|---|---|---|
| 8.48.0-4.b.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.c.1.10 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.d.1.9 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.d.2.5 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.e.1.11 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.e.2.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.h.1.10 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.i.1.12 | $8$ | $2$ | $2$ | $0$ |
| 8.48.1-8.c.1.4 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1-8.d.1.5 | $8$ | $2$ | $2$ | $1$ |
| 24.48.0-12.c.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.e.1.9 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.h.1.21 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.h.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.i.1.17 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.i.2.11 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.l.1.11 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.m.1.15 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-24.c.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.d.1.15 | $24$ | $2$ | $2$ | $1$ |
| 24.72.2-12.b.1.16 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-12.b.1.2 | $24$ | $4$ | $4$ | $1$ |
| 40.48.0-20.c.1.8 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.e.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.h.1.9 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.h.2.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.i.1.11 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.i.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.l.1.11 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.m.1.9 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1-40.c.1.15 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.d.1.5 | $40$ | $2$ | $2$ | $1$ |
| 40.120.4-20.b.1.11 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-20.b.1.26 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-20.b.1.21 | $40$ | $10$ | $10$ | $7$ |
| 56.48.0-28.c.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.e.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.h.1.13 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.h.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.i.1.11 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.i.2.9 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.l.1.11 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.m.1.11 | $56$ | $2$ | $2$ | $0$ |
| 56.48.1-56.c.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.d.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.192.5-28.b.1.10 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-28.b.1.15 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-28.b.1.26 | $56$ | $28$ | $28$ | $21$ |
| 88.48.0-44.c.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.e.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.h.1.13 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.h.2.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.i.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.i.2.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.l.1.11 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.m.1.11 | $88$ | $2$ | $2$ | $0$ |
| 88.48.1-88.c.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1-88.d.1.15 | $88$ | $2$ | $2$ | $1$ |
| 88.288.9-44.b.1.6 | $88$ | $12$ | $12$ | $9$ |
| 104.48.0-52.c.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.e.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.h.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.h.2.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.i.1.11 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.i.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.l.1.10 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.m.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.48.1-104.c.1.15 | $104$ | $2$ | $2$ | $1$ |
| 104.48.1-104.d.1.5 | $104$ | $2$ | $2$ | $1$ |
| 104.336.11-52.b.1.24 | $104$ | $14$ | $14$ | $11$ |
| 120.48.0-60.c.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.e.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.t.1.26 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.t.2.18 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.u.1.25 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.u.2.17 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.x.1.23 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.y.1.22 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-120.c.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.d.1.9 | $120$ | $2$ | $2$ | $1$ |
| 136.48.0-68.c.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.e.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.h.1.9 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.h.2.1 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.i.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.i.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.l.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.m.1.9 | $136$ | $2$ | $2$ | $0$ |
| 136.48.1-136.c.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1-136.d.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.432.15-68.b.1.10 | $136$ | $18$ | $18$ | $15$ |
| 152.48.0-76.c.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.e.1.9 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.h.1.13 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.h.2.2 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.i.1.10 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.i.2.9 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.l.1.11 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.m.1.11 | $152$ | $2$ | $2$ | $0$ |
| 152.48.1-152.c.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.48.1-152.d.1.15 | $152$ | $2$ | $2$ | $1$ |
| 152.480.17-76.b.1.15 | $152$ | $20$ | $20$ | $17$ |
| 168.48.0-84.c.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.e.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.t.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.t.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.u.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.u.2.19 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.x.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.y.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-168.c.1.30 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.d.1.6 | $168$ | $2$ | $2$ | $1$ |
| 184.48.0-92.c.1.6 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.e.1.9 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.h.1.13 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.h.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.i.1.11 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.i.2.9 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.l.1.11 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.m.1.11 | $184$ | $2$ | $2$ | $0$ |
| 184.48.1-184.c.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.48.1-184.d.1.15 | $184$ | $2$ | $2$ | $1$ |
| 232.48.0-116.c.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.e.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.h.1.9 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.h.2.1 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.i.1.11 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.i.2.3 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.l.1.11 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.m.1.9 | $232$ | $2$ | $2$ | $0$ |
| 232.48.1-232.c.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.48.1-232.d.1.5 | $232$ | $2$ | $2$ | $1$ |
| 248.48.0-124.c.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.e.1.9 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.h.1.13 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.h.2.3 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.i.1.11 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.i.2.9 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.l.1.11 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.m.1.11 | $248$ | $2$ | $2$ | $0$ |
| 248.48.1-248.c.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.48.1-248.d.1.15 | $248$ | $2$ | $2$ | $1$ |
| 264.48.0-132.c.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.e.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.t.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.t.2.25 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.u.1.25 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.u.2.18 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.x.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.y.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-264.c.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.d.1.29 | $264$ | $2$ | $2$ | $1$ |
| 280.48.0-140.c.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.e.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.t.1.18 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.t.2.26 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.u.1.17 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.u.2.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.x.1.20 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.y.1.20 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-280.c.1.15 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.d.1.17 | $280$ | $2$ | $2$ | $1$ |
| 296.48.0-148.c.1.5 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.e.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.h.1.9 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.h.2.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.i.1.11 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.i.2.3 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.l.1.11 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.m.1.9 | $296$ | $2$ | $2$ | $0$ |
| 296.48.1-296.c.1.8 | $296$ | $2$ | $2$ | $1$ |
| 296.48.1-296.d.1.5 | $296$ | $2$ | $2$ | $1$ |
| 312.48.0-156.c.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.e.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.t.1.17 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.t.2.17 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.u.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.u.2.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.x.1.22 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.y.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-312.c.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.d.1.9 | $312$ | $2$ | $2$ | $1$ |
| 328.48.0-164.c.1.6 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.e.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.h.1.9 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.h.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.i.1.11 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.i.2.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.l.1.10 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.m.1.9 | $328$ | $2$ | $2$ | $0$ |
| 328.48.1-328.c.1.8 | $328$ | $2$ | $2$ | $1$ |
| 328.48.1-328.d.1.5 | $328$ | $2$ | $2$ | $1$ |