Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-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 |
| Sutherland and Zywina (SZ) label: | 8J0-8a |
| Rouse and Zureick-Brown (RZB) label: | X99 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.15 |
Level structure
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
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\pm1}(2,4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.48.0.a.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.b.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.d.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.e.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.g.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.h.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.j.1 | $8$ | $2$ | $2$ | $0$ |
| 8.48.0.k.2 | $8$ | $2$ | $2$ | $0$ |
| 8.48.1.e.2 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1.i.1 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1.l.1 | $8$ | $2$ | $2$ | $1$ |
| 8.48.1.m.2 | $8$ | $2$ | $2$ | $1$ |
| 24.48.0.g.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.h.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.k.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.l.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.p.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.q.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.t.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0.u.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1.bc.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.bd.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.bg.2 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.bh.1 | $24$ | $2$ | $2$ | $1$ |
| 24.72.4.s.2 | $24$ | $3$ | $3$ | $4$ |
| 24.96.3.bn.2 | $24$ | $4$ | $4$ | $3$ |
| 40.48.0.i.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.j.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.m.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.n.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.q.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.r.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.u.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0.v.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1.bc.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.bd.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.bg.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1.bh.2 | $40$ | $2$ | $2$ | $1$ |
| 40.120.8.k.2 | $40$ | $5$ | $5$ | $8$ |
| 40.144.7.q.2 | $40$ | $6$ | $6$ | $7$ |
| 40.240.15.s.2 | $40$ | $10$ | $10$ | $15$ |
| 56.48.0.g.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.h.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.k.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.l.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.o.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.p.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.s.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.t.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.1.bc.2 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1.bd.2 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1.bg.2 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1.bh.1 | $56$ | $2$ | $2$ | $1$ |
| 56.192.11.p.2 | $56$ | $8$ | $8$ | $11$ |
| 56.504.34.s.2 | $56$ | $21$ | $21$ | $34$ |
| 56.672.45.u.1 | $56$ | $28$ | $28$ | $45$ |
| 88.48.0.g.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.h.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.k.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.l.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.o.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.p.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.s.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0.t.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.1.bc.2 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1.bd.2 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1.bg.2 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1.bh.1 | $88$ | $2$ | $2$ | $1$ |
| 88.288.19.x.2 | $88$ | $12$ | $12$ | $19$ |
| 104.48.0.i.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.j.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.m.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.n.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.q.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.r.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.u.2 | $104$ | $2$ | $2$ | $0$ |
| 104.48.0.v.1 | $104$ | $2$ | $2$ | $0$ |
| 104.48.1.bc.2 | $104$ | $2$ | $2$ | $1$ |
| 104.48.1.bd.2 | $104$ | $2$ | $2$ | $1$ |
| 104.48.1.bg.2 | $104$ | $2$ | $2$ | $1$ |
| 104.48.1.bh.2 | $104$ | $2$ | $2$ | $1$ |
| 104.336.23.q.1 | $104$ | $14$ | $14$ | $23$ |
| 120.48.0.z.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bb.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bh.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bj.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bp.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.br.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0.bz.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1.ds.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.du.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.ea.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.ec.2 | $120$ | $2$ | $2$ | $1$ |
| 136.48.0.i.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.j.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.m.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.n.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.q.1 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.r.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.u.2 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0.v.1 | $136$ | $2$ | $2$ | $0$ |
| 136.48.1.bc.2 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.bd.2 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.bg.2 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.bh.2 | $136$ | $2$ | $2$ | $1$ |
| 152.48.0.g.2 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.h.2 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.k.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.l.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.o.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.p.2 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.s.2 | $152$ | $2$ | $2$ | $0$ |
| 152.48.0.t.1 | $152$ | $2$ | $2$ | $0$ |
| 152.48.1.bc.2 | $152$ | $2$ | $2$ | $1$ |
| 152.48.1.bd.2 | $152$ | $2$ | $2$ | $1$ |
| 152.48.1.bg.2 | $152$ | $2$ | $2$ | $1$ |
| 152.48.1.bh.1 | $152$ | $2$ | $2$ | $1$ |
| 168.48.0.x.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.z.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bf.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bh.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bn.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bp.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bv.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.bx.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1.ds.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.du.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.ea.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.ec.2 | $168$ | $2$ | $2$ | $1$ |
| 184.48.0.g.2 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.h.2 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.k.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.l.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.o.2 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.p.2 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.s.2 | $184$ | $2$ | $2$ | $0$ |
| 184.48.0.t.1 | $184$ | $2$ | $2$ | $0$ |
| 184.48.1.bc.2 | $184$ | $2$ | $2$ | $1$ |
| 184.48.1.bd.2 | $184$ | $2$ | $2$ | $1$ |
| 184.48.1.bg.1 | $184$ | $2$ | $2$ | $1$ |
| 184.48.1.bh.1 | $184$ | $2$ | $2$ | $1$ |
| 232.48.0.i.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.j.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.m.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.n.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.q.1 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.r.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.u.2 | $232$ | $2$ | $2$ | $0$ |
| 232.48.0.v.1 | $232$ | $2$ | $2$ | $0$ |
| 232.48.1.bc.2 | $232$ | $2$ | $2$ | $1$ |
| 232.48.1.bd.2 | $232$ | $2$ | $2$ | $1$ |
| 232.48.1.bg.2 | $232$ | $2$ | $2$ | $1$ |
| 232.48.1.bh.2 | $232$ | $2$ | $2$ | $1$ |
| 248.48.0.g.2 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.h.2 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.k.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.l.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.o.2 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.p.2 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.s.2 | $248$ | $2$ | $2$ | $0$ |
| 248.48.0.t.1 | $248$ | $2$ | $2$ | $0$ |
| 248.48.1.bc.2 | $248$ | $2$ | $2$ | $1$ |
| 248.48.1.bd.2 | $248$ | $2$ | $2$ | $1$ |
| 248.48.1.bg.1 | $248$ | $2$ | $2$ | $1$ |
| 248.48.1.bh.1 | $248$ | $2$ | $2$ | $1$ |
| 264.48.0.x.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.z.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bf.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bh.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bn.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bp.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bv.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0.bx.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1.ds.2 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.du.2 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.ea.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.ec.2 | $264$ | $2$ | $2$ | $1$ |
| 280.48.0.y.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.ba.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.bg.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.bi.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.bo.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.bq.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.bw.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.by.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1.do.2 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.dq.2 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.dw.2 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1.dy.2 | $280$ | $2$ | $2$ | $1$ |
| 296.48.0.i.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.j.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.m.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.n.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.q.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.r.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.u.2 | $296$ | $2$ | $2$ | $0$ |
| 296.48.0.v.1 | $296$ | $2$ | $2$ | $0$ |
| 296.48.1.bc.2 | $296$ | $2$ | $2$ | $1$ |
| 296.48.1.bd.2 | $296$ | $2$ | $2$ | $1$ |
| 296.48.1.bg.2 | $296$ | $2$ | $2$ | $1$ |
| 296.48.1.bh.2 | $296$ | $2$ | $2$ | $1$ |
| 312.48.0.z.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bb.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bh.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bj.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bp.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.br.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bx.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0.bz.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1.ds.2 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.du.2 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.ea.2 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.ec.2 | $312$ | $2$ | $2$ | $1$ |
| 328.48.0.i.1 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.j.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.m.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.n.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.q.1 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.r.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.u.2 | $328$ | $2$ | $2$ | $0$ |
| 328.48.0.v.1 | $328$ | $2$ | $2$ | $0$ |
| 328.48.1.bc.2 | $328$ | $2$ | $2$ | $1$ |
| 328.48.1.bd.2 | $328$ | $2$ | $2$ | $1$ |
| 328.48.1.bg.2 | $328$ | $2$ | $2$ | $1$ |
| 328.48.1.bh.2 | $328$ | $2$ | $2$ | $1$ |