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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8J0 |
| Rouse and Zureick-Brown (RZB) label: | X98e |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.24 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\4&3\end{bmatrix}$, $\begin{bmatrix}1&0\\4&7\end{bmatrix}$, $\begin{bmatrix}5&2\\0&1\end{bmatrix}$, $\begin{bmatrix}5&2\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.e.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| 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 222 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{x^{24}(x^{8}-32x^{6}y^{2}+1280x^{4}y^{4}-16384x^{2}y^{6}+65536y^{8})^{3}}{y^{4}x^{32}(x-4y)^{4}(x+4y)^{4}(x^{2}-8y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(2,4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 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.b.2.9 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.e.2.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.f.1.5 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.h.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.k.2.4 | $8$ | $2$ | $2$ | $0$ |
| $X_1(2,8)$ | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.i.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.k.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.m.1.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.n.1.1 | $8$ | $2$ | $2$ | $1$ |
| 24.96.0-24.i.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.j.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.m.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.n.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.r.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.s.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.v.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.w.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.be.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bf.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bi.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bj.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.z.1.1 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.bq.1.2 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.k.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.l.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.o.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.p.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.s.2.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.t.2.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.w.2.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.x.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.be.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bf.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bi.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bj.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.n.1.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.v.1.4 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.z.1.13 | $40$ | $10$ | $10$ | $15$ |
| 56.96.0-56.i.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.j.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.q.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.r.2.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.u.2.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.v.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.be.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bf.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bi.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bj.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.s.1.10 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.z.1.1 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.bb.2.1 | $56$ | $28$ | $28$ | $45$ |
| 88.96.0-88.i.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.j.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.m.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.n.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.q.2.5 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.r.2.5 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.u.2.5 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.v.2.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.be.1.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bf.1.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bi.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bj.1.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.0-104.k.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.l.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.o.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.p.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.s.2.7 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.t.2.5 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.w.2.7 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.x.2.6 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.be.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bf.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bi.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bj.1.1 | $104$ | $2$ | $2$ | $1$ |
| 120.96.0-120.be.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bg.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bm.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bo.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bu.2.9 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.bw.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cc.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ce.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.dx.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dz.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ef.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.eh.1.1 | $120$ | $2$ | $2$ | $1$ |
| 136.96.0-136.k.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.l.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.o.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.p.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.s.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.t.2.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.w.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.x.2.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.be.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bf.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bi.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bj.1.1 | $136$ | $2$ | $2$ | $1$ |
| 152.96.0-152.i.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.j.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.m.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.n.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.q.2.5 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.r.2.5 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.u.2.5 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.v.2.6 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.be.1.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bf.1.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bi.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bj.1.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.0-168.bc.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.be.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bk.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bm.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bs.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bu.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ca.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cc.2.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.dx.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dz.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ef.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.eh.1.1 | $168$ | $2$ | $2$ | $1$ |
| 184.96.0-184.i.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.j.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.m.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.n.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.q.2.5 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.r.2.5 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.u.2.5 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.v.2.6 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.be.1.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bf.1.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bi.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bj.1.1 | $184$ | $2$ | $2$ | $1$ |
| 232.96.0-232.k.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.l.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.o.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.p.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.s.2.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.t.2.5 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.w.2.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.x.2.6 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.be.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bf.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bi.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bj.1.1 | $232$ | $2$ | $2$ | $1$ |
| 248.96.0-248.i.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.j.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.m.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.n.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.q.2.5 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.r.2.5 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.u.2.5 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.v.2.6 | $248$ | $2$ | $2$ | $0$ |
| 248.96.1-248.be.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bf.1.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bi.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bj.1.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.0-264.bc.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.be.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bk.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bm.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bs.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.bu.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ca.2.9 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cc.2.10 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.dx.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dz.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.ef.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.eh.1.1 | $264$ | $2$ | $2$ | $1$ |
| 280.96.0-280.bd.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bf.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bl.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bn.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bt.2.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bv.2.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cb.2.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cd.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.dt.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dv.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.eb.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ed.1.1 | $280$ | $2$ | $2$ | $1$ |
| 296.96.0-296.k.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.l.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.o.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.p.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.s.2.7 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.t.2.5 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.w.2.7 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.x.2.6 | $296$ | $2$ | $2$ | $0$ |
| 296.96.1-296.be.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bf.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bi.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bj.1.1 | $296$ | $2$ | $2$ | $1$ |
| 312.96.0-312.be.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bg.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bm.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bo.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bu.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.bw.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cc.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ce.2.10 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.dx.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dz.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ef.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.eh.1.1 | $312$ | $2$ | $2$ | $1$ |
| 328.96.0-328.k.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.l.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.o.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.p.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.s.2.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.t.2.5 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.w.2.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.x.2.6 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.be.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bf.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bi.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bj.1.1 | $328$ | $2$ | $2$ | $1$ |