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: | X98n |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.11 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\4&5\end{bmatrix}$, $\begin{bmatrix}5&2\\4&7\end{bmatrix}$, $\begin{bmatrix}7&0\\0&1\end{bmatrix}$, $\begin{bmatrix}7&0\\0&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-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: | $4$ |
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 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 |
---|---|---|---|---|---|
4.24.0-4.b.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-4.b.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.1.10 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.e.2.6 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.f.1.7 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.h.2.7 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.i.1.7 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.k.2.7 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.l.2.5 | $8$ | $2$ | $2$ | $0$ |
8.96.1-8.i.1.8 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.k.1.4 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.m.1.8 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.n.1.8 | $8$ | $2$ | $2$ | $1$ |
24.96.0-24.i.1.8 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.j.1.3 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.m.1.6 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.n.1.8 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.r.2.12 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.s.2.10 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.v.2.10 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.w.2.15 | $24$ | $2$ | $2$ | $0$ |
24.96.1-24.be.1.7 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bf.1.15 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bi.1.16 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bj.1.12 | $24$ | $2$ | $2$ | $1$ |
24.144.4-24.z.1.22 | $24$ | $3$ | $3$ | $4$ |
24.192.3-24.bq.1.55 | $24$ | $4$ | $4$ | $3$ |
40.96.0-40.k.1.8 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.l.1.2 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.o.1.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.p.1.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.s.2.10 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.t.2.12 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.w.2.14 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.x.2.11 | $40$ | $2$ | $2$ | $0$ |
40.96.1-40.be.1.6 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bf.1.8 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bi.1.4 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bj.1.12 | $40$ | $2$ | $2$ | $1$ |
40.240.8-40.n.1.24 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.v.1.58 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.z.1.64 | $40$ | $10$ | $10$ | $15$ |
56.96.0-56.i.1.8 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.j.1.2 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.m.1.4 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.n.1.8 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.q.2.14 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.r.2.12 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.u.2.12 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.v.2.15 | $56$ | $2$ | $2$ | $0$ |
56.96.1-56.be.1.7 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bf.1.15 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bi.1.16 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bj.1.8 | $56$ | $2$ | $2$ | $1$ |
56.384.11-56.s.1.53 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.z.1.64 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.bb.2.64 | $56$ | $28$ | $28$ | $45$ |
88.96.0-88.i.1.8 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.j.1.3 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.m.1.6 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.n.1.8 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.q.2.12 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.r.2.10 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.u.2.10 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.v.2.15 | $88$ | $2$ | $2$ | $0$ |
88.96.1-88.be.1.7 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bf.1.15 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bi.1.16 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bj.1.12 | $88$ | $2$ | $2$ | $1$ |
104.96.0-104.k.1.8 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.l.1.2 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.o.1.4 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.p.1.4 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.s.2.12 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.t.2.12 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.w.2.10 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.x.2.11 | $104$ | $2$ | $2$ | $0$ |
104.96.1-104.be.1.6 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bf.1.8 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bi.1.4 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bj.1.12 | $104$ | $2$ | $2$ | $1$ |
120.96.0-120.be.1.2 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bg.1.8 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bm.1.4 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bo.1.4 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bu.2.21 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bw.2.30 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.cc.2.32 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.ce.2.28 | $120$ | $2$ | $2$ | $0$ |
120.96.1-120.dx.1.16 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.dz.1.4 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.ef.1.8 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.eh.1.8 | $120$ | $2$ | $2$ | $1$ |
136.96.0-136.k.1.8 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.l.1.5 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.o.1.6 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.p.1.6 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.s.2.14 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.t.2.10 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.w.2.13 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.x.2.15 | $136$ | $2$ | $2$ | $0$ |
136.96.1-136.be.1.10 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bf.1.12 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bi.1.10 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bj.1.12 | $136$ | $2$ | $2$ | $1$ |
152.96.0-152.i.1.8 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.j.1.3 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.m.1.6 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.n.1.8 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.q.2.12 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.r.2.10 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.u.2.10 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.v.2.15 | $152$ | $2$ | $2$ | $0$ |
152.96.1-152.be.1.7 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bf.1.15 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bi.1.16 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bj.1.12 | $152$ | $2$ | $2$ | $1$ |
168.96.0-168.bc.1.8 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.be.1.6 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bk.1.12 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bm.1.4 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bs.2.28 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bu.2.22 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.ca.2.29 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cc.2.31 | $168$ | $2$ | $2$ | $0$ |
168.96.1-168.dx.1.12 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.dz.1.16 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ef.1.8 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.eh.1.24 | $168$ | $2$ | $2$ | $1$ |
184.96.0-184.i.1.8 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.j.1.3 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.m.1.7 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.n.1.8 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.q.2.12 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.r.2.10 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.u.2.10 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.v.2.15 | $184$ | $2$ | $2$ | $0$ |
184.96.1-184.be.1.11 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bf.1.15 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bi.1.16 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bj.1.14 | $184$ | $2$ | $2$ | $1$ |
232.96.0-232.k.1.8 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.l.1.5 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.o.1.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.p.1.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.s.2.14 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.t.2.12 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.w.2.14 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.x.2.15 | $232$ | $2$ | $2$ | $0$ |
232.96.1-232.be.1.10 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bf.1.12 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bi.1.10 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bj.1.12 | $232$ | $2$ | $2$ | $1$ |
248.96.0-248.i.1.8 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.j.1.3 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.m.1.7 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.n.1.8 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.q.2.12 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.r.2.10 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.u.2.10 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.v.2.15 | $248$ | $2$ | $2$ | $0$ |
248.96.1-248.be.1.11 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bf.1.15 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bi.1.16 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bj.1.14 | $248$ | $2$ | $2$ | $1$ |
264.96.0-264.bc.1.8 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.be.1.6 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bk.1.14 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bm.1.8 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bs.2.26 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bu.2.22 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.ca.2.22 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.cc.2.31 | $264$ | $2$ | $2$ | $0$ |
264.96.1-264.dx.1.23 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.dz.1.15 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.ef.1.16 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.eh.1.28 | $264$ | $2$ | $2$ | $1$ |
280.96.0-280.bd.1.4 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bf.1.4 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bl.1.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bn.1.8 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bt.2.20 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bv.2.29 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.cb.2.29 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.cd.2.32 | $280$ | $2$ | $2$ | $0$ |
280.96.1-280.dt.1.8 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.dv.1.8 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.eb.1.16 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.ed.1.10 | $280$ | $2$ | $2$ | $1$ |
296.96.0-296.k.1.8 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.l.1.2 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.o.1.4 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.p.1.4 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.s.2.14 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.t.2.10 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.w.2.13 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.x.2.15 | $296$ | $2$ | $2$ | $0$ |
296.96.1-296.be.1.4 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bf.1.12 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bi.1.6 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bj.1.8 | $296$ | $2$ | $2$ | $1$ |
312.96.0-312.be.1.8 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bg.1.6 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bm.1.8 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bo.1.6 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bu.2.32 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bw.2.22 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.cc.2.21 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.ce.2.31 | $312$ | $2$ | $2$ | $0$ |
312.96.1-312.dx.1.12 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.dz.1.16 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.ef.1.12 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.eh.1.16 | $312$ | $2$ | $2$ | $1$ |
328.96.0-328.k.1.8 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.l.1.2 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.o.1.4 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.p.1.4 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.s.2.16 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.t.2.10 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.w.2.9 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.x.2.15 | $328$ | $2$ | $2$ | $0$ |
328.96.1-328.be.1.4 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bf.1.12 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bi.1.6 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bj.1.8 | $328$ | $2$ | $2$ | $1$ |