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: | X60c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.4 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\0&3\end{bmatrix}$, $\begin{bmatrix}3&1\\4&1\end{bmatrix}$, $\begin{bmatrix}5&4\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\wr C_2$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.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 41 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{(2x+y)^{24}(16x^{2}+32xy+13y^{2})^{3}(48x^{2}+48xy+11y^{2})^{3}(3328x^{4}+7936x^{3}y+7136x^{2}y^{2}+2864xy^{3}+433y^{4})^{3}}{(2x+y)^{28}(4x+3y)^{4}(16x^{2}+16xy+5y^{2})^{4}(80x^{2}+96xy+29y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.16.0-4.b.1.1 | $8$ | $3$ | $3$ | $0$ | $0$ |
| 8.24.0-4.d.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.d.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 |
|---|---|---|---|---|
| 8.96.1-8.z.1.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.bb.1.4 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.bk.1.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.bm.1.2 | $8$ | $2$ | $2$ | $1$ |
| 24.96.1-24.dy.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.ec.1.3 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.eu.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.ey.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-12.p.1.8 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-12.m.1.10 | $24$ | $4$ | $4$ | $3$ |
| 40.96.1-40.dg.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.dk.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.dw.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.ea.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-20.f.1.4 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-20.p.1.10 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-20.p.1.1 | $40$ | $10$ | $10$ | $15$ |
| 56.96.1-56.dg.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.dk.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.dw.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.ea.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-28.m.1.1 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-28.p.1.3 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-28.t.1.4 | $56$ | $28$ | $28$ | $45$ |
| 88.96.1-88.dg.1.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.dk.1.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.dw.1.4 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.ea.1.3 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1-104.dg.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.dk.1.4 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.dw.1.3 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.ea.1.2 | $104$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ky.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.lg.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.me.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.mm.1.2 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1-136.dg.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.dk.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.dw.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.ea.1.2 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1-152.dg.1.3 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.dk.1.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.dw.1.4 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.ea.1.4 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ky.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.lg.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.me.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.mm.1.6 | $168$ | $2$ | $2$ | $1$ |
| 184.96.1-184.dg.1.3 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.dk.1.3 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.dw.1.3 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.ea.1.3 | $184$ | $2$ | $2$ | $1$ |
| 232.96.1-232.dg.1.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.dk.1.3 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.dw.1.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.ea.1.2 | $232$ | $2$ | $2$ | $1$ |
| 248.96.1-248.dg.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.dk.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.dw.1.3 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.ea.1.3 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ky.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.lg.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.me.1.5 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.mm.1.3 | $264$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ka.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ki.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.lg.1.8 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.lo.1.2 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1-296.dg.1.2 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.dk.1.4 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.dw.1.3 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.ea.1.2 | $296$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ky.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.lg.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.me.1.6 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.mm.1.6 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1-328.dg.1.2 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.dk.1.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.dw.1.3 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.ea.1.2 | $328$ | $2$ | $2$ | $1$ |