Invariants
| Level: | $8$ | $\SL_2$-level: | $4$ | ||||
| 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 $2$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse and Zureick-Brown (RZB) label: | X62e |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.8 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\4&7\end{bmatrix}$, $\begin{bmatrix}3&2\\4&1\end{bmatrix}$, $\begin{bmatrix}7&6\\0&5\end{bmatrix}$, $\begin{bmatrix}7&6\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.c.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 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(x+y)^{24}(x^{8}+56x^{4}y^{4}+16y^{8})^{3}}{y^{4}x^{4}(x+y)^{24}(x^{2}-2y^{2})^{4}(x^{2}+2y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.b.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.a.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.b.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.b.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.b.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.d.1.6 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.e.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.e.2.6 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.f.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.j.1.4 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.p.1.3 | $8$ | $2$ | $2$ | $1$ |
| 24.96.0-24.d.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.e.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.e.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.f.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.v.1.5 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bb.1.9 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.c.1.4 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.f.1.18 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.d.1.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.e.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.e.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.f.1.8 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.v.1.7 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bb.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.c.1.12 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.c.1.1 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.c.1.8 | $40$ | $10$ | $10$ | $15$ |
| 56.96.0-56.d.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.e.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.e.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.f.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.v.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bb.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.c.1.14 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.c.1.22 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.c.1.16 | $56$ | $28$ | $28$ | $45$ |
| 88.96.0-88.d.1.8 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.e.1.8 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.e.2.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.f.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.v.1.5 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bb.1.5 | $88$ | $2$ | $2$ | $1$ |
| 104.96.0-104.d.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.e.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.e.2.6 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.f.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.v.1.7 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bb.1.1 | $104$ | $2$ | $2$ | $1$ |
| 120.96.0-120.d.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.e.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.e.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.f.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.v.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bb.1.3 | $120$ | $2$ | $2$ | $1$ |
| 136.96.0-136.d.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.e.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.e.2.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.f.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.v.1.7 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bb.1.1 | $136$ | $2$ | $2$ | $1$ |
| 152.96.0-152.d.1.8 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.e.1.8 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.e.2.6 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.f.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.v.1.5 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bb.1.5 | $152$ | $2$ | $2$ | $1$ |
| 168.96.0-168.d.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.e.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.e.2.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.f.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.v.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bb.1.2 | $168$ | $2$ | $2$ | $1$ |
| 184.96.0-184.d.1.8 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.e.1.8 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.e.2.6 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.f.1.6 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.v.1.5 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bb.1.5 | $184$ | $2$ | $2$ | $1$ |
| 232.96.0-232.d.1.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.e.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.e.2.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.f.1.8 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.v.1.7 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bb.1.1 | $232$ | $2$ | $2$ | $1$ |
| 248.96.0-248.d.1.8 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.e.1.8 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.e.2.6 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.f.1.6 | $248$ | $2$ | $2$ | $0$ |
| 248.96.1-248.v.1.5 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bb.1.5 | $248$ | $2$ | $2$ | $1$ |
| 264.96.0-264.d.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.e.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.e.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.f.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.v.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bb.1.10 | $264$ | $2$ | $2$ | $1$ |
| 280.96.0-280.d.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.e.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.e.2.16 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.f.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.v.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.bb.1.5 | $280$ | $2$ | $2$ | $1$ |
| 296.96.0-296.d.1.6 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.e.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.e.2.6 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.f.1.8 | $296$ | $2$ | $2$ | $0$ |
| 296.96.1-296.v.1.7 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bb.1.1 | $296$ | $2$ | $2$ | $1$ |
| 312.96.0-312.d.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.e.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.e.2.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.f.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.v.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bb.1.1 | $312$ | $2$ | $2$ | $1$ |
| 328.96.0-328.d.1.6 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.e.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.e.2.6 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.f.1.8 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.v.1.7 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bb.1.1 | $328$ | $2$ | $2$ | $1$ |