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 | $1^{2}\cdot2\cdot8$ | 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: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
| Rouse and Zureick-Brown (RZB) label: | X34g |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.103 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&1\\4&7\end{bmatrix}$, $\begin{bmatrix}3&6\\4&7\end{bmatrix}$, $\begin{bmatrix}5&3\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_4:D_4$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.p.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| 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 1543 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{12}(x^{2}-4xy-8y^{2})^{3}(x^{2}+4xy-8y^{2})^{3}}{y^{8}x^{14}(x^{2}-32y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.h.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.k.1.6 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.x.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.y.1.3 | $8$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bp.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.br.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bt.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bv.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-24.cx.1.9 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-24.ix.1.22 | $24$ | $4$ | $4$ | $1$ |
| 40.48.0-40.bt.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bv.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bx.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bz.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-40.br.1.13 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-40.ch.1.27 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-40.cx.1.19 | $40$ | $10$ | $10$ | $7$ |
| 56.48.0-56.bn.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bp.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.br.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bt.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.192.5-56.br.1.2 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.cx.1.7 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.cx.1.25 | $56$ | $28$ | $28$ | $21$ |
| 88.48.0-88.bn.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bp.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.br.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bt.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-88.br.1.13 | $88$ | $12$ | $12$ | $9$ |
| 104.48.0-104.bt.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bv.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bx.1.5 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bz.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-104.ch.1.7 | $104$ | $14$ | $14$ | $11$ |
| 120.48.0-120.dx.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dz.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ef.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eh.1.5 | $120$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bt.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bv.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bx.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bz.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.432.15-136.ct.1.6 | $136$ | $18$ | $18$ | $15$ |
| 152.48.0-152.bn.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bp.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.br.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bt.1.5 | $152$ | $2$ | $2$ | $0$ |
| 152.480.17-152.br.1.7 | $152$ | $20$ | $20$ | $17$ |
| 168.48.0-168.dr.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dt.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dz.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.eb.1.6 | $168$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bn.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bp.1.5 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.br.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bt.1.5 | $184$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bt.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bv.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bx.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bz.1.6 | $232$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bn.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bp.1.5 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.br.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bt.1.5 | $248$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dr.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dt.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dz.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.eb.1.5 | $264$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dx.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dz.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ef.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.eh.1.9 | $280$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bt.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bv.1.5 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bx.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bz.1.7 | $296$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dx.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dz.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ef.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.eh.1.7 | $312$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bt.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bv.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bx.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bz.1.3 | $328$ | $2$ | $2$ | $0$ |