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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
| Rouse and Zureick-Brown (RZB) label: | X99h |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.31 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\0&5\end{bmatrix}$, $\begin{bmatrix}3&4\\4&5\end{bmatrix}$, $\begin{bmatrix}5&6\\4&1\end{bmatrix}$, $\begin{bmatrix}7&6\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.d.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| 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 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.b.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.b.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.a.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.b.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.d.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.e.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.g.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.h.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.j.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.k.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.e.2.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.i.1.5 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.l.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.m.2.1 | $8$ | $2$ | $2$ | $1$ |
| 24.96.0-24.g.2.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.h.2.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.k.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.l.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.p.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.q.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.t.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.u.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.bc.2.10 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bd.2.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bg.2.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bh.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.s.2.1 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.bn.2.2 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.i.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.j.2.2 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.m.2.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.n.2.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.q.2.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.r.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.u.2.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.v.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.bc.2.15 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bd.2.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bg.2.9 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bh.2.13 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.k.2.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.q.2.13 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.s.2.1 | $40$ | $10$ | $10$ | $15$ |
| 56.96.0-56.g.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.h.2.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.k.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.o.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.p.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.s.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.t.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.bc.2.10 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bd.2.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bg.2.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bh.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.p.2.11 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.s.2.4 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.u.1.1 | $56$ | $28$ | $28$ | $45$ |
| 88.96.0-88.g.2.7 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.h.2.2 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.k.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.l.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.o.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.p.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.s.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.t.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.bc.2.10 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bd.2.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bg.2.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bh.1.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.0-104.i.1.7 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.j.2.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.m.2.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.n.2.5 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.q.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.r.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.u.2.4 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.v.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.bc.2.15 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bd.2.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bg.2.9 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bh.2.13 | $104$ | $2$ | $2$ | $1$ |
| 120.96.0-120.z.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bb.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bh.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bj.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bp.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.br.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bx.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bz.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.ds.2.17 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.du.2.29 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ea.2.25 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ec.2.25 | $120$ | $2$ | $2$ | $1$ |
| 136.96.0-136.i.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.j.2.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.m.2.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.n.2.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.q.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.r.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.u.2.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.v.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.bc.2.15 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bd.2.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bg.2.9 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bh.2.13 | $136$ | $2$ | $2$ | $1$ |
| 152.96.0-152.g.2.7 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.h.2.2 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.k.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.l.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.o.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.p.2.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.s.2.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.t.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.bc.2.10 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bd.2.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bg.2.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bh.1.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.0-168.x.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.z.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bf.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bh.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bn.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bp.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bv.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bx.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.ds.2.25 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.du.2.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ea.2.25 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ec.2.17 | $168$ | $2$ | $2$ | $1$ |
| 184.96.0-184.g.2.7 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.h.2.2 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.k.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.l.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.o.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.p.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.s.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.t.1.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.bc.2.10 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bd.2.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bg.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bh.1.1 | $184$ | $2$ | $2$ | $1$ |
| 232.96.0-232.i.2.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.j.2.2 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.m.2.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.n.2.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.q.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.r.2.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.u.2.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.v.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.bc.2.15 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bd.2.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bg.2.9 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bh.2.13 | $232$ | $2$ | $2$ | $1$ |
| 248.96.0-248.g.2.7 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.h.2.2 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.k.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.l.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.o.2.3 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.p.2.3 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.s.2.3 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.t.1.3 | $248$ | $2$ | $2$ | $0$ |
| 248.96.1-248.bc.2.10 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bd.2.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bg.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bh.1.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.0-264.x.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.z.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bf.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bh.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bn.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bp.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bv.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bx.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.ds.2.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.du.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ea.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ec.2.1 | $264$ | $2$ | $2$ | $1$ |
| 280.96.0-280.y.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ba.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bg.2.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bi.2.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bo.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bq.2.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bw.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.by.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.do.2.25 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dq.2.25 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dw.2.17 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dy.2.29 | $280$ | $2$ | $2$ | $1$ |
| 296.96.0-296.i.1.7 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.j.2.2 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.m.2.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.n.2.5 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.q.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.r.2.3 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.u.2.4 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.v.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.96.1-296.bc.2.15 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bd.2.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bg.2.9 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bh.2.13 | $296$ | $2$ | $2$ | $1$ |
| 312.96.0-312.z.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bb.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bh.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bj.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bp.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.br.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bx.2.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bz.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.ds.2.25 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.du.2.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ea.2.25 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ec.2.17 | $312$ | $2$ | $2$ | $1$ |
| 328.96.0-328.i.1.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.j.2.2 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.m.2.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.n.2.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.q.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.r.2.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.u.2.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.v.1.4 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.bc.2.15 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bd.2.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bg.2.9 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bh.2.13 | $328$ | $2$ | $2$ | $1$ |