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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse and Zureick-Brown (RZB) label: | X27c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.73 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&7\\4&3\end{bmatrix}$, $\begin{bmatrix}3&1\\0&5\end{bmatrix}$, $\begin{bmatrix}5&0\\4&1\end{bmatrix}$, $\begin{bmatrix}5&5\\4&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_4^2:C_2^2$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.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 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{(4x-y)^{12}(64x^{2}-32xy+y^{2})^{3}(64x^{2}+32xy+y^{2})^{3}}{y^{2}x^{2}(4x-y)^{12}(64x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.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 |
|---|---|---|---|---|
| 8.48.0-4.c.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.p.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.q.1.5 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.t.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.w.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0-8.x.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.1-8.m.1.4 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1-8.n.1.4 | $8$ | $2$ | $2$ | $1$ |
| 24.48.0-12.e.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-12.p.1.14 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-12.h.1.23 | $24$ | $4$ | $4$ | $1$ |
| 40.48.0-20.d.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-20.h.1.6 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-20.l.1.9 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-20.p.1.14 | $40$ | $10$ | $10$ | $7$ |
| 24.48.0-24.u.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.v.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.z.1.7 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bc.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bd.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-24.m.1.6 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.n.1.4 | $24$ | $2$ | $2$ | $1$ |
| 56.48.0-28.d.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.192.5-28.h.1.13 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-28.p.1.22 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-28.p.1.21 | $56$ | $28$ | $28$ | $21$ |
| 40.48.0-40.v.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.w.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.z.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.be.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bf.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1-40.m.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.n.1.2 | $40$ | $2$ | $2$ | $1$ |
| 88.48.0-44.d.1.5 | $88$ | $2$ | $2$ | $0$ |
| 88.288.9-44.h.1.9 | $88$ | $12$ | $12$ | $9$ |
| 104.48.0-52.d.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.336.11-52.l.1.5 | $104$ | $14$ | $14$ | $11$ |
| 56.48.0-56.t.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.u.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.x.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.ba.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bb.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.1-56.m.1.6 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.n.1.6 | $56$ | $2$ | $2$ | $1$ |
| 120.48.0-60.h.1.10 | $120$ | $2$ | $2$ | $0$ |
| 136.48.0-68.d.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.432.15-68.l.1.11 | $136$ | $18$ | $18$ | $15$ |
| 152.48.0-76.d.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.480.17-76.h.1.13 | $152$ | $20$ | $20$ | $17$ |
| 168.48.0-84.h.1.6 | $168$ | $2$ | $2$ | $0$ |
| 88.48.0-88.t.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.u.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.x.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.ba.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.bb.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.48.1-88.m.1.6 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1-88.n.1.4 | $88$ | $2$ | $2$ | $1$ |
| 184.48.0-92.d.1.6 | $184$ | $2$ | $2$ | $0$ |
| 104.48.0-104.v.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.w.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.z.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.be.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0-104.bf.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.48.1-104.m.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.48.1-104.n.1.2 | $104$ | $2$ | $2$ | $1$ |
| 232.48.0-116.d.1.6 | $232$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bh.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bi.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bl.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bq.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.br.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-120.m.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.n.1.2 | $120$ | $2$ | $2$ | $1$ |
| 248.48.0-124.d.1.6 | $248$ | $2$ | $2$ | $0$ |
| 264.48.0-132.h.1.8 | $264$ | $2$ | $2$ | $0$ |
| 136.48.0-136.t.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.u.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.z.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.be.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.bf.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.1-136.m.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1-136.n.1.2 | $136$ | $2$ | $2$ | $1$ |
| 280.48.0-140.h.1.10 | $280$ | $2$ | $2$ | $0$ |
| 296.48.0-148.d.1.6 | $296$ | $2$ | $2$ | $0$ |
| 152.48.0-152.t.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.u.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.x.1.7 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.ba.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0-152.bb.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.48.1-152.m.1.6 | $152$ | $2$ | $2$ | $1$ |
| 152.48.1-152.n.1.4 | $152$ | $2$ | $2$ | $1$ |
| 312.48.0-156.h.1.11 | $312$ | $2$ | $2$ | $0$ |
| 328.48.0-164.d.1.3 | $328$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bf.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bg.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bj.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bm.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bn.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-168.m.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.n.1.2 | $168$ | $2$ | $2$ | $1$ |
| 184.48.0-184.t.1.3 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.u.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.x.1.6 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.ba.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0-184.bb.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.48.1-184.m.1.6 | $184$ | $2$ | $2$ | $1$ |
| 184.48.1-184.n.1.4 | $184$ | $2$ | $2$ | $1$ |
| 232.48.0-232.v.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.w.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.z.1.5 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.be.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0-232.bf.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.48.1-232.m.1.3 | $232$ | $2$ | $2$ | $1$ |
| 232.48.1-232.n.1.2 | $232$ | $2$ | $2$ | $1$ |
| 248.48.0-248.t.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.u.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.x.1.7 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.ba.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0-248.bb.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.48.1-248.m.1.6 | $248$ | $2$ | $2$ | $1$ |
| 248.48.1-248.n.1.4 | $248$ | $2$ | $2$ | $1$ |
| 264.48.0-264.bf.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bg.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bj.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bm.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bn.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-264.m.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.n.1.6 | $264$ | $2$ | $2$ | $1$ |
| 280.48.0-280.bh.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bi.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bl.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bq.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.br.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-280.m.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.n.1.2 | $280$ | $2$ | $2$ | $1$ |
| 296.48.0-296.v.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.w.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.z.1.5 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.be.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0-296.bf.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.48.1-296.m.1.3 | $296$ | $2$ | $2$ | $1$ |
| 296.48.1-296.n.1.2 | $296$ | $2$ | $2$ | $1$ |
| 312.48.0-312.bh.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bi.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bl.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.bq.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.br.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-312.m.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.n.1.2 | $312$ | $2$ | $2$ | $1$ |
| 328.48.0-328.t.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.u.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.z.1.5 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.be.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0-328.bf.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.48.1-328.m.1.3 | $328$ | $2$ | $2$ | $1$ |
| 328.48.1-328.n.1.2 | $328$ | $2$ | $2$ | $1$ |