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 $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 4G0 |
| Rouse and Zureick-Brown (RZB) label: | X58k |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.47 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}3&0\\0&1\end{bmatrix}$, $\begin{bmatrix}3&0\\0&3\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}5&4\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^5$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.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: | $32$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.b.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.96.0-8.a.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.a.1.6 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.1.11 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.2.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.2.6 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.2.10 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.c.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.c.1.7 | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.g.1.3 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.g.1.7 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.g.1.10 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.g.2.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.1.7 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.1.12 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.2.3 | $8$ | $2$ | $2$ | $1$ |
| 8.96.2-8.a.1.3 | $8$ | $2$ | $2$ | $2$ |
| 8.96.2-8.a.1.8 | $8$ | $2$ | $2$ | $2$ |
| 24.96.0-24.a.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.a.1.17 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.1.9 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.2.11 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.c.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.c.1.18 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.n.1.5 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.n.2.6 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.n.2.10 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.n.2.19 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.1.5 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.2.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.2.16 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.2.23 | $24$ | $2$ | $2$ | $1$ |
| 24.96.2-24.a.1.12 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.a.1.14 | $24$ | $2$ | $2$ | $2$ |
| 24.144.4-12.b.1.5 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-12.b.1.1 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.a.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.a.1.17 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.1.8 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.1.9 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.2.11 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.c.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.c.1.17 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.n.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.2.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.2.9 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.2.22 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.2.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.2.15 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.2.23 | $40$ | $2$ | $2$ | $1$ |
| 40.96.2-40.a.1.9 | $40$ | $2$ | $2$ | $2$ |
| 40.96.2-40.a.1.15 | $40$ | $2$ | $2$ | $2$ |
| 40.240.8-20.b.1.11 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-20.b.1.8 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-20.b.1.9 | $40$ | $10$ | $10$ | $15$ |
| 56.96.0-56.a.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.a.1.17 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.1.9 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.2.11 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.c.1.9 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.c.1.18 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.n.1.4 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.n.2.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.n.2.9 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.n.2.18 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.2.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.2.15 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.2.23 | $56$ | $2$ | $2$ | $1$ |
| 56.96.2-56.a.1.10 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.a.1.16 | $56$ | $2$ | $2$ | $2$ |
| 56.384.11-28.b.1.26 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-28.b.1.12 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-28.b.1.9 | $56$ | $28$ | $28$ | $45$ |
| 88.96.0-88.a.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.a.1.17 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.8 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.11 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.c.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.c.1.18 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.n.1.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.6 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.9 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.20 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.1.5 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.15 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.24 | $88$ | $2$ | $2$ | $1$ |
| 88.96.2-88.a.1.12 | $88$ | $2$ | $2$ | $2$ |
| 88.96.2-88.a.1.14 | $88$ | $2$ | $2$ | $2$ |
| 104.96.0-104.a.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.a.1.17 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.6 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.11 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.c.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.c.1.18 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.n.1.3 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.4 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.9 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.24 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.1.4 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.10 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.15 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.23 | $104$ | $2$ | $2$ | $1$ |
| 104.96.2-104.a.1.9 | $104$ | $2$ | $2$ | $2$ |
| 104.96.2-104.a.1.15 | $104$ | $2$ | $2$ | $2$ |
| 120.96.0-120.a.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.a.1.34 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.1.22 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.2.23 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.c.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.c.1.39 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.n.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.2.30 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.2.45 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.2.30 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.2.45 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.a.1.10 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.a.1.16 | $120$ | $2$ | $2$ | $2$ |
| 136.96.0-136.a.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.a.1.17 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.9 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.18 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.n.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.9 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.21 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.15 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.23 | $136$ | $2$ | $2$ | $1$ |
| 136.96.2-136.a.1.11 | $136$ | $2$ | $2$ | $2$ |
| 136.96.2-136.a.1.13 | $136$ | $2$ | $2$ | $2$ |
| 152.96.0-152.a.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.a.1.17 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.1.5 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.1.8 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.1.9 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.2.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.2.6 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.2.11 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.c.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.c.1.18 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.n.1.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.n.2.6 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.n.2.9 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.n.2.22 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.1.5 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.2.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.2.15 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.2.24 | $152$ | $2$ | $2$ | $1$ |
| 152.96.2-152.a.1.12 | $152$ | $2$ | $2$ | $2$ |
| 152.96.2-152.a.1.14 | $152$ | $2$ | $2$ | $2$ |
| 168.96.0-168.a.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.a.1.35 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.23 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.c.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.c.1.33 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.n.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.36 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.29 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.48 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-168.a.1.19 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.29 | $168$ | $2$ | $2$ | $2$ |
| 184.96.0-184.a.1.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.a.1.17 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.1.5 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.1.8 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.1.9 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.2.6 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.2.11 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.c.1.7 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.c.1.18 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.n.1.4 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.n.2.6 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.n.2.9 | $184$ | $2$ | $2$ | $1$ |
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