Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| 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 | $1^{2}\cdot2\cdot4\cdot8^{2}$ | 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: | 8I0 |
| Rouse and Zureick-Brown (RZB) label: | X84l |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.175 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\0&5\end{bmatrix}$, $\begin{bmatrix}5&3\\0&1\end{bmatrix}$, $\begin{bmatrix}7&1\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_8:C_2$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| Cyclic 8-torsion field degree: | $2$ |
| 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 138 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{x^{24}(x^{8}+4x^{6}y^{2}-10x^{4}y^{4}-28x^{2}y^{6}+y^{8})^{3}}{y^{4}x^{26}(x^{2}+y^{2})^{8}(x^{2}+2y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.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 |
|---|---|---|---|---|
| 8.96.0-8.j.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.m.2.4 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.n.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.o.1.2 | $8$ | $2$ | $2$ | $0$ |
| 16.96.0-16.u.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.w.1.1 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.y.1.1 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.ba.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.1-16.q.1.7 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.s.1.8 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.u.1.8 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.w.1.7 | $16$ | $2$ | $2$ | $1$ |
| 24.96.0-24.bi.2.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bk.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bm.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bo.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.144.4-24.ge.1.5 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.gf.2.3 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.bi.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bk.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bm.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bo.2.4 | $40$ | $2$ | $2$ | $0$ |
| 40.240.8-40.da.2.7 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.fn.2.9 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.gq.2.1 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.be.1.7 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bg.1.5 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bm.2.13 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bo.1.15 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.bq.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bs.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.by.1.12 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.ca.1.10 | $48$ | $2$ | $2$ | $1$ |
| 56.96.0-56.bg.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bi.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bk.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bm.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.384.11-56.fb.2.3 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.ge.1.6 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.gq.2.5 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.bm.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.bo.2.2 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.bu.2.4 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.bw.2.8 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.bs.2.9 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.bu.2.13 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.ca.2.15 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.cc.1.13 | $80$ | $2$ | $2$ | $1$ |
| 88.96.0-88.bg.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bi.1.2 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bk.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bm.1.2 | $88$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bi.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bk.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bm.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bo.2.4 | $104$ | $2$ | $2$ | $0$ |
| 112.96.0-112.be.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bg.2.2 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bm.2.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bo.2.8 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bq.2.9 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bs.2.13 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.by.2.15 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.ca.1.13 | $112$ | $2$ | $2$ | $1$ |
| 120.96.0-120.ed.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.eh.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.el.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ep.1.2 | $120$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bi.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bk.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bm.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bo.1.2 | $136$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bg.2.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bi.1.2 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bk.2.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bm.1.2 | $152$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dz.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ed.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eh.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.el.1.4 | $168$ | $2$ | $2$ | $0$ |
| 176.96.0-176.be.1.4 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.bg.1.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.bm.2.4 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.bo.2.8 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.bq.2.9 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.bs.2.13 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.by.1.15 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.ca.1.13 | $176$ | $2$ | $2$ | $1$ |
| 184.96.0-184.bg.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bi.1.2 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bk.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bm.1.2 | $184$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bm.1.4 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bo.1.2 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bu.2.4 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.bw.2.8 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-208.bs.2.9 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.bu.2.13 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.ca.1.15 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.cc.1.13 | $208$ | $2$ | $2$ | $1$ |
| 232.96.0-232.bi.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bk.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bm.1.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bo.1.4 | $232$ | $2$ | $2$ | $0$ |
| 240.96.0-240.co.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cq.2.3 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.de.2.7 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.dg.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.fo.1.17 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fq.2.18 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ge.2.26 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gg.1.25 | $240$ | $2$ | $2$ | $1$ |
| 248.96.0-248.bg.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bi.1.2 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bk.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bm.1.2 | $248$ | $2$ | $2$ | $0$ |
| 264.96.0-264.dz.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ed.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.eh.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.el.1.2 | $264$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bm.2.4 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bo.2.8 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bu.1.4 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bw.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.bs.1.15 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bu.1.13 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.ca.2.9 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.cc.2.13 | $272$ | $2$ | $2$ | $1$ |
| 280.96.0-280.dw.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ea.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ee.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ei.1.2 | $280$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bi.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bk.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bm.1.2 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bo.2.4 | $296$ | $2$ | $2$ | $0$ |
| 304.96.0-304.be.1.4 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bg.1.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bm.2.4 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.bo.2.8 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.bq.2.9 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.bs.2.13 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.by.1.15 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.ca.1.13 | $304$ | $2$ | $2$ | $1$ |
| 312.96.0-312.ed.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.eh.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.el.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ep.1.4 | $312$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bi.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bk.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bm.1.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bo.1.2 | $328$ | $2$ | $2$ | $0$ |