Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
| Rouse and Zureick-Brown (RZB) label: | X48a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.118 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}1&1\\4&1\end{bmatrix}$, $\begin{bmatrix}7&2\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_4:D_4$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.d.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 624 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2}\cdot\frac{(8x-y)^{12}(1024x^{4}-448x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(8x-y)^{12}(32x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.48.0-8.j.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.j.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.k.1.5 | $8$ | $2$ | $2$ | $0$ |
| 24.48.0-24.n.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.n.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.o.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.o.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-24.j.1.13 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-24.cg.1.17 | $24$ | $4$ | $4$ | $1$ |
| 40.48.0-40.n.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.n.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.o.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.o.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-40.f.1.5 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-40.h.1.9 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-40.j.1.11 | $40$ | $10$ | $10$ | $7$ |
| 56.48.0-56.n.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.n.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.o.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.o.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.192.5-56.f.1.13 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.j.1.16 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.j.1.9 | $56$ | $28$ | $28$ | $21$ |
| 88.48.0-88.n.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.n.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.o.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.o.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-88.f.1.13 | $88$ | $12$ | $12$ | $9$ |
| 104.48.0-104.n.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.n.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.o.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.o.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.h.1.9 | $104$ | $14$ | $14$ | $11$ |
| 120.48.0-120.z.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.z.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ba.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ba.1.5 | $120$ | $2$ | $2$ | $0$ |
| 136.48.0-136.n.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.n.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.o.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.o.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.432.15-136.h.1.10 | $136$ | $18$ | $18$ | $15$ |
| 152.48.0-152.n.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.n.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.o.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.o.1.5 | $152$ | $2$ | $2$ | $0$ |
| 152.480.17-152.f.1.13 | $152$ | $20$ | $20$ | $17$ |
| 168.48.0-168.z.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.z.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ba.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ba.1.5 | $168$ | $2$ | $2$ | $0$ |
| 184.48.0-184.n.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.n.1.3 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.o.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.o.1.3 | $184$ | $2$ | $2$ | $0$ |
| 232.48.0-232.n.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.n.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.o.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.o.1.3 | $232$ | $2$ | $2$ | $0$ |
| 248.48.0-248.n.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.n.1.3 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.o.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.o.1.3 | $248$ | $2$ | $2$ | $0$ |
| 264.48.0-264.z.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.z.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ba.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ba.1.5 | $264$ | $2$ | $2$ | $0$ |
| 280.48.0-280.z.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.z.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ba.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ba.1.5 | $280$ | $2$ | $2$ | $0$ |
| 296.48.0-296.n.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.n.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.o.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.o.1.3 | $296$ | $2$ | $2$ | $0$ |
| 312.48.0-312.z.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.z.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ba.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ba.1.5 | $312$ | $2$ | $2$ | $0$ |
| 328.48.0-328.n.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.n.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.o.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.o.1.5 | $328$ | $2$ | $2$ | $0$ |