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: | X101h |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.113 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\0&7\end{bmatrix}$, $\begin{bmatrix}3&4\\4&7\end{bmatrix}$, $\begin{bmatrix}5&0\\0&7\end{bmatrix}$, $\begin{bmatrix}7&6\\0&7\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
Contains $-I$: | no $\quad$ (see 8.24.0.e.1 for the level structure with $-I$) |
Cyclic 8-isogeny field degree: | $2$ |
Cyclic 8-torsion field degree: | $8$ |
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 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-4.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-4.b.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.1.2 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.c.1.2 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.e.1.8 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.f.1.4 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.h.1.4 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.i.1.6 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.k.1.7 | $8$ | $2$ | $2$ | $0$ |
8.96.0-8.l.1.4 | $8$ | $2$ | $2$ | $0$ |
8.96.1-8.i.2.2 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.k.2.2 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.m.2.5 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.n.1.3 | $8$ | $2$ | $2$ | $1$ |
24.96.0-24.i.2.4 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.j.2.10 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.m.2.14 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.n.2.16 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.r.1.15 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.s.1.16 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.v.1.16 | $24$ | $2$ | $2$ | $0$ |
24.96.0-24.w.1.14 | $24$ | $2$ | $2$ | $0$ |
24.96.1-24.be.2.11 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bf.2.4 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bi.2.10 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.bj.2.10 | $24$ | $2$ | $2$ | $1$ |
24.144.4-24.z.2.47 | $24$ | $3$ | $3$ | $4$ |
24.192.3-24.bq.2.47 | $24$ | $4$ | $4$ | $3$ |
40.96.0-40.k.2.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.l.2.14 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.o.2.14 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.p.2.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.s.1.15 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.t.1.11 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.w.1.16 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.x.1.14 | $40$ | $2$ | $2$ | $0$ |
40.96.1-40.be.2.13 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bf.2.4 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bi.2.6 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bj.2.9 | $40$ | $2$ | $2$ | $1$ |
40.240.8-40.n.2.12 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.v.2.37 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.z.2.23 | $40$ | $10$ | $10$ | $15$ |
56.96.0-56.i.2.4 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.j.2.10 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.m.2.12 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.n.2.14 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.q.1.14 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.r.1.12 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.u.1.12 | $56$ | $2$ | $2$ | $0$ |
56.96.0-56.v.1.12 | $56$ | $2$ | $2$ | $0$ |
56.96.1-56.be.2.11 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bf.2.4 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bi.2.10 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.bj.2.10 | $56$ | $2$ | $2$ | $1$ |
56.384.11-56.s.2.42 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.z.2.27 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.bb.1.45 | $56$ | $28$ | $28$ | $45$ |
88.96.0-88.i.2.4 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.j.2.14 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.m.2.14 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.n.2.16 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.q.1.15 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.r.1.16 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.u.1.16 | $88$ | $2$ | $2$ | $0$ |
88.96.0-88.v.1.12 | $88$ | $2$ | $2$ | $0$ |
88.96.1-88.be.2.11 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bf.2.4 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bi.2.10 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.bj.2.10 | $88$ | $2$ | $2$ | $1$ |
104.96.0-104.k.2.4 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.l.2.14 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.o.2.16 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.p.2.4 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.s.1.16 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.t.1.12 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.w.1.10 | $104$ | $2$ | $2$ | $0$ |
104.96.0-104.x.1.14 | $104$ | $2$ | $2$ | $0$ |
104.96.1-104.be.2.13 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bf.2.4 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bi.2.6 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.bj.2.9 | $104$ | $2$ | $2$ | $1$ |
120.96.0-120.be.2.18 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bg.2.3 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bm.2.3 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bo.2.20 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bu.1.18 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bw.1.19 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.cc.1.18 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.ce.1.19 | $120$ | $2$ | $2$ | $0$ |
120.96.1-120.dx.2.2 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.dz.2.17 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.ef.2.2 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.eh.2.25 | $120$ | $2$ | $2$ | $1$ |
136.96.0-136.k.2.4 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.l.2.10 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.o.2.10 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.p.2.6 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.s.1.11 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.t.1.10 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.w.1.16 | $136$ | $2$ | $2$ | $0$ |
136.96.0-136.x.1.10 | $136$ | $2$ | $2$ | $0$ |
136.96.1-136.be.2.13 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bf.2.4 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bi.2.10 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.bj.2.9 | $136$ | $2$ | $2$ | $1$ |
152.96.0-152.i.2.4 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.j.2.10 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.m.2.14 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.n.2.16 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.q.1.15 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.r.1.16 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.u.1.16 | $152$ | $2$ | $2$ | $0$ |
152.96.0-152.v.1.12 | $152$ | $2$ | $2$ | $0$ |
152.96.1-152.be.2.11 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bf.2.4 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bi.2.10 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.bj.2.10 | $152$ | $2$ | $2$ | $1$ |
168.96.0-168.bc.2.4 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.be.2.20 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bk.2.20 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bm.2.4 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bs.1.19 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.bu.1.19 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.ca.1.30 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cc.1.18 | $168$ | $2$ | $2$ | $0$ |
168.96.1-168.dx.2.26 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.dz.2.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.ef.2.18 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.eh.2.18 | $168$ | $2$ | $2$ | $1$ |
184.96.0-184.i.2.4 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.j.2.14 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.m.2.14 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.n.2.16 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.q.1.15 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.r.1.16 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.u.1.16 | $184$ | $2$ | $2$ | $0$ |
184.96.0-184.v.1.12 | $184$ | $2$ | $2$ | $0$ |
184.96.1-184.be.2.11 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bf.2.4 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bi.2.10 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.bj.2.10 | $184$ | $2$ | $2$ | $1$ |
232.96.0-232.k.2.4 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.l.2.10 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.o.2.10 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.p.2.6 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.s.1.11 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.t.1.10 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.w.1.16 | $232$ | $2$ | $2$ | $0$ |
232.96.0-232.x.1.10 | $232$ | $2$ | $2$ | $0$ |
232.96.1-232.be.2.13 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bf.2.4 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bi.2.10 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.bj.2.9 | $232$ | $2$ | $2$ | $1$ |
248.96.0-248.i.2.4 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.j.2.14 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.m.2.14 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.n.2.16 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.q.1.15 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.r.1.16 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.u.1.16 | $248$ | $2$ | $2$ | $0$ |
248.96.0-248.v.1.12 | $248$ | $2$ | $2$ | $0$ |
248.96.1-248.be.2.11 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bf.2.4 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bi.2.10 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.bj.2.10 | $248$ | $2$ | $2$ | $1$ |
264.96.0-264.bc.2.4 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.be.2.20 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bk.2.28 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bm.2.24 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bs.1.30 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.bu.1.23 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.ca.1.27 | $264$ | $2$ | $2$ | $0$ |
264.96.0-264.cc.1.20 | $264$ | $2$ | $2$ | $0$ |
264.96.1-264.dx.2.20 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.dz.2.6 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.ef.2.18 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.eh.2.18 | $264$ | $2$ | $2$ | $1$ |
280.96.0-280.bd.2.17 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bf.2.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bl.2.3 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bn.2.21 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bt.1.18 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bv.1.22 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.cb.1.18 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.cd.1.20 | $280$ | $2$ | $2$ | $0$ |
280.96.1-280.dt.2.2 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.dv.2.17 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.eb.2.4 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.ed.2.25 | $280$ | $2$ | $2$ | $1$ |
296.96.0-296.k.2.4 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.l.2.10 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.o.2.10 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.p.2.4 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.s.1.11 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.t.1.10 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.w.1.16 | $296$ | $2$ | $2$ | $0$ |
296.96.0-296.x.1.10 | $296$ | $2$ | $2$ | $0$ |
296.96.1-296.be.2.13 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bf.2.4 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bi.2.10 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.bj.2.9 | $296$ | $2$ | $2$ | $1$ |
312.96.0-312.be.2.4 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bg.2.20 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bm.2.24 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bo.2.6 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bu.1.20 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.bw.1.20 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.cc.1.18 | $312$ | $2$ | $2$ | $0$ |
312.96.0-312.ce.1.18 | $312$ | $2$ | $2$ | $0$ |
312.96.1-312.dx.2.26 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.dz.2.4 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.ef.2.18 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.eh.2.18 | $312$ | $2$ | $2$ | $1$ |
328.96.0-328.k.2.4 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.l.2.10 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.o.2.12 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.p.2.4 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.s.1.12 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.t.1.12 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.w.1.10 | $328$ | $2$ | $2$ | $0$ |
328.96.0-328.x.1.10 | $328$ | $2$ | $2$ | $0$ |
328.96.1-328.be.2.13 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bf.2.4 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bi.2.10 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.bj.2.9 | $328$ | $2$ | $2$ | $1$ |