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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
| Rouse and Zureick-Brown (RZB) label: | X96o |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.35 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\0&1\end{bmatrix}$, $\begin{bmatrix}3&4\\0&1\end{bmatrix}$, $\begin{bmatrix}5&2\\0&5\end{bmatrix}$, $\begin{bmatrix}5&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.i.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| 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 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.24.0-4.b.1.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-4.b.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.o.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
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.j.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.j.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.k.2.2 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.l.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.96.0-8.l.2.4 | $8$ | $2$ | $2$ | $0$ |
| 8.96.1-8.h.1.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1-8.p.1.2 | $8$ | $2$ | $2$ | $1$ |
| 16.96.0-16.d.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.d.2.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.1-16.a.1.4 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.a.2.4 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.b.2.2 | $16$ | $2$ | $2$ | $1$ |
| 16.96.2-16.d.1.10 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2-16.d.2.9 | $16$ | $2$ | $2$ | $2$ |
| 24.96.0-24.ba.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.ba.2.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bb.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bb.2.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bc.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bc.2.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.bu.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bv.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.ch.1.38 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.cl.1.18 | $24$ | $4$ | $4$ | $3$ |
| 40.96.0-40.bb.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bb.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bc.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bc.2.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bd.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bd.2.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.bu.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bv.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.v.1.24 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.br.1.29 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.ch.1.1 | $40$ | $10$ | $10$ | $15$ |
| 48.96.0-48.d.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.d.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.a.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.a.2.6 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.b.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.b.2.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.2-48.d.1.6 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.d.2.5 | $48$ | $2$ | $2$ | $2$ |
| 56.96.0-56.z.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.z.2.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.ba.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.ba.2.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bb.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bb.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.bu.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bv.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.bn.1.7 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.ch.1.14 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.cj.1.3 | $56$ | $28$ | $28$ | $45$ |
| 80.96.0-80.f.1.1 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.f.2.1 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.a.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.a.2.5 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.b.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.b.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.96.2-80.f.1.9 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-80.f.2.5 | $80$ | $2$ | $2$ | $2$ |
| 88.96.0-88.z.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.z.2.5 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.ba.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.ba.2.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bb.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.bb.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-88.bu.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.bv.1.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.0-104.bb.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bb.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bc.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bc.2.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bd.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.bd.2.7 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-104.bu.1.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.bv.1.3 | $104$ | $2$ | $2$ | $1$ |
| 112.96.0-112.d.1.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.d.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.a.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.a.2.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.b.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.b.2.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.2-112.d.1.11 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2-112.d.2.11 | $112$ | $2$ | $2$ | $2$ |
| 120.96.0-120.cy.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cy.2.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cz.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cz.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.da.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.da.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.fq.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fr.1.3 | $120$ | $2$ | $2$ | $1$ |
| 136.96.0-136.bb.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bb.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bd.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bd.2.7 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.bu.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bv.1.3 | $136$ | $2$ | $2$ | $1$ |
| 152.96.0-152.z.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.z.2.5 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.ba.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.ba.2.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bb.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.96.0-152.bb.2.3 | $152$ | $2$ | $2$ | $0$ |
| 152.96.1-152.bu.1.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1-152.bv.1.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.0-168.cw.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cw.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cx.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cx.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cy.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cy.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.fq.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fr.1.6 | $168$ | $2$ | $2$ | $1$ |
| 176.96.0-176.d.1.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.d.2.2 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.a.1.4 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.a.2.4 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.b.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.b.2.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.2-176.d.1.20 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2-176.d.2.11 | $176$ | $2$ | $2$ | $2$ |
| 184.96.0-184.z.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.z.2.5 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.ba.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.ba.2.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bb.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.96.0-184.bb.2.3 | $184$ | $2$ | $2$ | $0$ |
| 184.96.1-184.bu.1.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1-184.bv.1.1 | $184$ | $2$ | $2$ | $1$ |
| 208.96.0-208.f.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.f.2.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-208.a.1.5 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.a.2.5 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.b.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.b.2.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.2-208.f.1.5 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2-208.f.2.5 | $208$ | $2$ | $2$ | $2$ |
| 232.96.0-232.bb.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bb.2.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bc.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bc.2.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bd.1.3 | $232$ | $2$ | $2$ | $0$ |
| 232.96.0-232.bd.2.7 | $232$ | $2$ | $2$ | $0$ |
| 232.96.1-232.bu.1.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1-232.bv.1.3 | $232$ | $2$ | $2$ | $1$ |
| 240.96.0-240.f.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.f.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.a.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.2.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-240.f.1.19 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.2.11 | $240$ | $2$ | $2$ | $2$ |
| 248.96.0-248.z.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.z.2.5 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.ba.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.ba.2.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bb.1.1 | $248$ | $2$ | $2$ | $0$ |
| 248.96.0-248.bb.2.3 | $248$ | $2$ | $2$ | $0$ |
| 248.96.1-248.bu.1.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1-248.bv.1.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.0-264.cw.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cw.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cx.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cx.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cy.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cy.2.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.fq.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fr.1.2 | $264$ | $2$ | $2$ | $1$ |
| 272.96.0-272.f.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.f.2.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.a.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.a.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.b.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.b.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.96.2-272.f.1.12 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.f.2.12 | $272$ | $2$ | $2$ | $2$ |
| 280.96.0-280.cx.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cx.2.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cy.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cy.2.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cz.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cz.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.fm.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fn.1.5 | $280$ | $2$ | $2$ | $1$ |
| 296.96.0-296.bb.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bb.2.3 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bc.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bc.2.3 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bd.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.96.0-296.bd.2.7 | $296$ | $2$ | $2$ | $0$ |
| 296.96.1-296.bu.1.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1-296.bv.1.3 | $296$ | $2$ | $2$ | $1$ |
| 304.96.0-304.d.1.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.d.2.2 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.a.1.4 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.a.2.4 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.b.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.b.2.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.2-304.d.1.12 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2-304.d.2.11 | $304$ | $2$ | $2$ | $2$ |
| 312.96.0-312.cy.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cy.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cz.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cz.2.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.da.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.da.2.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.fq.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fr.1.5 | $312$ | $2$ | $2$ | $1$ |
| 328.96.0-328.bb.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bb.2.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bc.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bc.2.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bd.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.96.0-328.bd.2.7 | $328$ | $2$ | $2$ | $0$ |
| 328.96.1-328.bu.1.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1-328.bv.1.3 | $328$ | $2$ | $2$ | $1$ |