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: | X58j |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.54 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}1&0\\4&5\end{bmatrix}$, $\begin{bmatrix}1&4\\4&1\end{bmatrix}$, $\begin{bmatrix}3&4\\0&3\end{bmatrix}$, $\begin{bmatrix}5&4\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-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.a.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.a.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.1.7 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.2.5 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.c.1.8 | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.g.1.6 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.g.2.3 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.1.8 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.h.2.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.2-8.a.1.5 | $8$ | $2$ | $2$ | $2$ |
| 8.96.2-8.a.1.10 | $8$ | $2$ | $2$ | $2$ |
| 24.96.0-24.a.1.19 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.b.2.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.c.1.15 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.n.1.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.n.2.24 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.1.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.o.2.22 | $24$ | $2$ | $2$ | $1$ |
| 24.96.2-24.a.1.7 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.a.1.24 | $24$ | $2$ | $2$ | $2$ |
| 24.144.4-12.b.1.7 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-12.b.1.27 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.a.1.19 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.2.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.c.1.16 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.n.1.5 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.2.23 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.2.22 | $40$ | $2$ | $2$ | $1$ |
| 40.96.2-40.a.1.8 | $40$ | $2$ | $2$ | $2$ |
| 40.96.2-40.a.1.23 | $40$ | $2$ | $2$ | $2$ |
| 40.240.8-20.b.1.18 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-20.b.1.13 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-20.b.1.17 | $40$ | $10$ | $10$ | $15$ |
| 56.96.0-56.a.1.19 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.b.2.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.c.1.15 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.n.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.n.2.21 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.o.2.20 | $56$ | $2$ | $2$ | $1$ |
| 56.96.2-56.a.1.7 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.a.1.23 | $56$ | $2$ | $2$ | $2$ |
| 56.384.11-28.b.1.35 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-28.b.1.14 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-28.b.1.16 | $56$ | $28$ | $28$ | $45$ |
| 88.96.0-88.a.1.19 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.7 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.c.1.15 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.n.1.5 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.23 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.1.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.21 | $88$ | $2$ | $2$ | $1$ |
| 88.96.2-88.a.1.7 | $88$ | $2$ | $2$ | $2$ |
| 88.96.2-88.a.1.23 | $88$ | $2$ | $2$ | $2$ |
| 104.96.0-104.a.1.19 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.7 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.c.1.15 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.n.1.5 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.21 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.1.3 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.20 | $104$ | $2$ | $2$ | $1$ |
| 104.96.2-104.a.1.8 | $104$ | $2$ | $2$ | $2$ |
| 104.96.2-104.a.1.23 | $104$ | $2$ | $2$ | $2$ |
| 120.96.0-120.a.1.39 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.2.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.c.1.38 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.n.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.2.44 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.2.48 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.a.1.7 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.a.1.46 | $120$ | $2$ | $2$ | $2$ |
| 136.96.0-136.a.1.19 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.15 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.n.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.24 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.22 | $136$ | $2$ | $2$ | $1$ |
| 136.96.2-136.a.1.8 | $136$ | $2$ | $2$ | $2$ |
| 136.96.2-136.a.1.23 | $136$ | $2$ | $2$ | $2$ |
| 152.96.0-152.a.1.19 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.b.2.7 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.c.1.15 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.n.1.5 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.n.2.17 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.1.3 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.o.2.21 | $152$ | $2$ | $2$ | $1$ |
| 152.96.2-152.a.1.7 | $152$ | $2$ | $2$ | $2$ |
| 152.96.2-152.a.1.23 | $152$ | $2$ | $2$ | $2$ |
| 168.96.0-168.a.1.37 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.b.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.c.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.n.1.10 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.n.2.41 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.2.41 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-168.a.1.16 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.a.1.46 | $168$ | $2$ | $2$ | $2$ |
| 184.96.0-184.a.1.19 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.b.2.7 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.c.1.15 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.n.1.5 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.n.2.24 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.o.1.3 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.o.2.21 | $184$ | $2$ | $2$ | $1$ |
| 184.96.2-184.a.1.7 | $184$ | $2$ | $2$ | $2$ |
| 184.96.2-184.a.1.23 | $184$ | $2$ | $2$ | $2$ |
| 232.96.0-232.a.1.19 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.b.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.b.2.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.c.1.16 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.n.1.5 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.n.2.24 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.o.1.3 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.o.2.22 | $232$ | $2$ | $2$ | $1$ |
| 232.96.2-232.a.1.8 | $232$ | $2$ | $2$ | $2$ |
| 232.96.2-232.a.1.23 | $232$ | $2$ | $2$ | $2$ |
| 248.96.0-248.a.1.19 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.b.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.b.2.7 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.c.1.15 | $248$ | $2$ | $2$ | $0$ |
| 248.96.1-248.n.1.5 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.n.2.18 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.o.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.o.2.21 | $248$ | $2$ | $2$ | $1$ |
| 248.96.2-248.a.1.7 | $248$ | $2$ | $2$ | $2$ |
| 248.96.2-248.a.1.23 | $248$ | $2$ | $2$ | $2$ |
| 264.96.0-264.a.1.37 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.c.1.31 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.n.1.10 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.41 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.37 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-264.a.1.16 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.a.1.46 | $264$ | $2$ | $2$ | $2$ |
| 280.96.0-280.a.1.38 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.b.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.b.2.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.c.1.36 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.n.1.12 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.n.2.34 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.o.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.o.2.42 | $280$ | $2$ | $2$ | $1$ |
| 280.96.2-280.a.1.15 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.a.1.42 | $280$ | $2$ | $2$ | $2$ |
| 296.96.0-296.a.1.19 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.b.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.b.2.7 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.c.1.15 | $296$ | $2$ | $2$ | $0$ |
| 296.96.1-296.n.1.5 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.n.2.23 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.o.1.3 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.o.2.21 | $296$ | $2$ | $2$ | $1$ |
| 296.96.2-296.a.1.8 | $296$ | $2$ | $2$ | $2$ |
| 296.96.2-296.a.1.23 | $296$ | $2$ | $2$ | $2$ |
| 312.96.0-312.a.1.37 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.c.1.31 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.n.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.n.2.40 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.o.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.o.2.40 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-312.a.1.15 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.a.1.46 | $312$ | $2$ | $2$ | $2$ |
| 328.96.0-328.a.1.19 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.b.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.b.2.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.c.1.16 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.n.1.5 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.n.2.19 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.o.1.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.o.2.19 | $328$ | $2$ | $2$ | $1$ |
| 328.96.2-328.a.1.8 | $328$ | $2$ | $2$ | $2$ |
| 328.96.2-328.a.1.23 | $328$ | $2$ | $2$ | $2$ |