Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| 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: | 8N0 |
| Sutherland and Zywina (SZ) label: | 8N0-8f |
| Rouse and Zureick-Brown (RZB) label: | X189 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.45 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{48}(x^{8}-4x^{7}y+4x^{6}y^{2}+28x^{5}y^{3}+6x^{4}y^{4}-28x^{3}y^{5}+4x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+4x^{7}y+4x^{6}y^{2}-28x^{5}y^{3}+6x^{4}y^{4}+28x^{3}y^{5}+4x^{2}y^{6}-4xy^{7}+y^{8})^{3}}{y^{4}x^{52}(x-y)^{4}(x+y)^{4}(x^{2}+y^{2})^{8}(x^{2}-2xy-y^{2})^{4}(x^{2}+2xy-y^{2})^{4}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0.e.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0.e.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.96.1.f.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.f.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.g.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.g.2 | $8$ | $2$ | $2$ | $1$ |
| 8.96.3.i.1 | $8$ | $2$ | $2$ | $3$ |
| 8.96.3.j.1 | $8$ | $2$ | $2$ | $3$ |
| 16.96.2.a.1 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2.b.1 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2.c.1 | $16$ | $2$ | $2$ | $2$ |
| 16.96.2.d.1 | $16$ | $2$ | $2$ | $2$ |
| 24.96.1.w.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.w.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.x.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.x.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.3.z.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.ba.1 | $24$ | $2$ | $2$ | $3$ |
| 24.144.8.l.1 | $24$ | $3$ | $3$ | $8$ |
| 24.192.7.i.1 | $24$ | $4$ | $4$ | $7$ |
| 40.96.1.w.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.w.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.x.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.x.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.3.be.1 | $40$ | $2$ | $2$ | $3$ |
| 40.96.3.bf.1 | $40$ | $2$ | $2$ | $3$ |
| 40.240.16.f.1 | $40$ | $5$ | $5$ | $16$ |
| 40.288.15.i.1 | $40$ | $6$ | $6$ | $15$ |
| 40.480.31.l.1 | $40$ | $10$ | $10$ | $31$ |
| 48.96.2.a.1 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2.b.1 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2.c.1 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2.d.1 | $48$ | $2$ | $2$ | $2$ |
| 56.96.1.w.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.w.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.x.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.x.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.3.w.1 | $56$ | $2$ | $2$ | $3$ |
| 56.96.3.x.1 | $56$ | $2$ | $2$ | $3$ |
| 56.384.23.i.1 | $56$ | $8$ | $8$ | $23$ |
| 56.1008.70.l.1 | $56$ | $21$ | $21$ | $70$ |
| 56.1344.93.l.1 | $56$ | $28$ | $28$ | $93$ |
| 80.96.2.e.1 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2.f.1 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2.g.1 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2.h.1 | $80$ | $2$ | $2$ | $2$ |
| 88.96.1.w.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.w.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.x.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.x.2 | $88$ | $2$ | $2$ | $1$ |
| 88.96.3.w.1 | $88$ | $2$ | $2$ | $3$ |
| 88.96.3.x.1 | $88$ | $2$ | $2$ | $3$ |
| 104.96.1.w.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.w.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.x.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.x.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.3.be.1 | $104$ | $2$ | $2$ | $3$ |
| 104.96.3.bf.1 | $104$ | $2$ | $2$ | $3$ |
| 112.96.2.a.1 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2.b.1 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2.c.1 | $112$ | $2$ | $2$ | $2$ |
| 112.96.2.d.1 | $112$ | $2$ | $2$ | $2$ |
| 120.96.1.cy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.cy.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.cz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.cz.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.cw.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.cx.1 | $120$ | $2$ | $2$ | $3$ |
| 136.96.1.w.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.w.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.x.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.x.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.3.be.1 | $136$ | $2$ | $2$ | $3$ |
| 136.96.3.bf.1 | $136$ | $2$ | $2$ | $3$ |
| 152.96.1.w.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.w.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.x.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.x.2 | $152$ | $2$ | $2$ | $1$ |
| 152.96.3.w.1 | $152$ | $2$ | $2$ | $3$ |
| 152.96.3.x.1 | $152$ | $2$ | $2$ | $3$ |
| 168.96.1.cy.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.cy.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.cz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.cz.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.3.co.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.cp.1 | $168$ | $2$ | $2$ | $3$ |
| 176.96.2.a.1 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2.b.1 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2.c.1 | $176$ | $2$ | $2$ | $2$ |
| 176.96.2.d.1 | $176$ | $2$ | $2$ | $2$ |
| 184.96.1.w.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.w.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.x.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.x.2 | $184$ | $2$ | $2$ | $1$ |
| 184.96.3.w.1 | $184$ | $2$ | $2$ | $3$ |
| 184.96.3.x.1 | $184$ | $2$ | $2$ | $3$ |
| 208.96.2.c.1 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2.d.1 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2.i.1 | $208$ | $2$ | $2$ | $2$ |
| 208.96.2.j.1 | $208$ | $2$ | $2$ | $2$ |
| 232.96.1.w.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.w.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.x.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.x.2 | $232$ | $2$ | $2$ | $1$ |
| 232.96.3.be.1 | $232$ | $2$ | $2$ | $3$ |
| 232.96.3.bf.1 | $232$ | $2$ | $2$ | $3$ |
| 240.96.2.e.1 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2.f.1 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2.g.1 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2.h.1 | $240$ | $2$ | $2$ | $2$ |
| 248.96.1.w.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.w.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.x.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.x.2 | $248$ | $2$ | $2$ | $1$ |
| 248.96.3.w.1 | $248$ | $2$ | $2$ | $3$ |
| 248.96.3.x.1 | $248$ | $2$ | $2$ | $3$ |
| 264.96.1.cy.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.cy.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.cz.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.cz.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.3.co.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.cp.1 | $264$ | $2$ | $2$ | $3$ |
| 272.96.2.a.1 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2.b.1 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2.k.1 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2.l.1 | $272$ | $2$ | $2$ | $2$ |
| 280.96.1.cy.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.cy.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.cz.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.cz.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.3.cw.1 | $280$ | $2$ | $2$ | $3$ |
| 280.96.3.cx.1 | $280$ | $2$ | $2$ | $3$ |
| 296.96.1.w.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.w.2 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.x.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.x.2 | $296$ | $2$ | $2$ | $1$ |
| 296.96.3.be.1 | $296$ | $2$ | $2$ | $3$ |
| 296.96.3.bf.1 | $296$ | $2$ | $2$ | $3$ |
| 304.96.2.a.1 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2.b.1 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2.c.1 | $304$ | $2$ | $2$ | $2$ |
| 304.96.2.d.1 | $304$ | $2$ | $2$ | $2$ |
| 312.96.1.cy.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.cy.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.cz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.cz.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.3.cw.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.cx.1 | $312$ | $2$ | $2$ | $3$ |
| 328.96.1.w.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.w.2 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.x.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.x.2 | $328$ | $2$ | $2$ | $1$ |
| 328.96.3.be.1 | $328$ | $2$ | $2$ | $3$ |
| 328.96.3.bf.1 | $328$ | $2$ | $2$ | $3$ |