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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Sutherland and Zywina (SZ) label: | 8O0-8i |
| Rouse and Zureick-Brown (RZB) label: | X190 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.138 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(x-y)^{48}(x^{8}-4x^{7}y+16x^{6}y^{2}-56x^{5}y^{3}+120x^{4}y^{4}-112x^{3}y^{5}+64x^{2}y^{6}-32xy^{7}+16y^{8})^{3}(13x^{8}-140x^{7}y+688x^{6}y^{2}-1960x^{5}y^{3}+3480x^{4}y^{4}-3920x^{3}y^{5}+2752x^{2}y^{6}-1120xy^{7}+208y^{8})^{3}}{(x-y)^{48}(x^{2}-2y^{2})^{4}(x^{2}-4xy+2y^{2})^{8}(x^{2}-4xy+6y^{2})^{2}(x^{2}-2xy+2y^{2})^{8}(3x^{2}-4xy+2y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| 8.24.0.d.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0.e.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0.h.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 8.96.1.a.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.c.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.f.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.h.1 | $8$ | $2$ | $2$ | $1$ |
| 24.96.1.bu.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bv.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bw.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bx.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.8.ez.2 | $24$ | $3$ | $3$ | $8$ |
| 24.192.7.dg.2 | $24$ | $4$ | $4$ | $7$ |
| 40.96.1.bu.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bv.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bw.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bx.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.16.bh.2 | $40$ | $5$ | $5$ | $16$ |
| 40.288.15.dn.2 | $40$ | $6$ | $6$ | $15$ |
| 40.480.31.fb.2 | $40$ | $10$ | $10$ | $31$ |
| 56.96.1.bu.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bv.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bw.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bx.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.23.dg.2 | $56$ | $8$ | $8$ | $23$ |
| 56.1008.70.ez.2 | $56$ | $21$ | $21$ | $70$ |
| 56.1344.93.ez.1 | $56$ | $28$ | $28$ | $93$ |
| 88.96.1.bu.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.bv.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.bw.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.bx.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1.bu.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.bv.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.bw.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.bx.1 | $104$ | $2$ | $2$ | $1$ |
| 120.96.1.pg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ph.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.pi.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.pj.1 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1.bu.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.bv.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.bw.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.bx.1 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1.bu.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.bv.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.bw.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.bx.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1.pg.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ph.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.pi.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.pj.1 | $168$ | $2$ | $2$ | $1$ |
| 184.96.1.bu.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.bv.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.bw.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.bx.1 | $184$ | $2$ | $2$ | $1$ |
| 232.96.1.bu.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.bv.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.bw.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.bx.1 | $232$ | $2$ | $2$ | $1$ |
| 248.96.1.bu.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.bv.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.bw.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.bx.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1.pg.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ph.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.pi.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.pj.1 | $264$ | $2$ | $2$ | $1$ |
| 280.96.1.om.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.on.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.oo.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.op.1 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1.bu.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.bv.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.bw.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.bx.1 | $296$ | $2$ | $2$ | $1$ |
| 312.96.1.pg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ph.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.pi.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.pj.1 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1.bu.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.bv.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.bw.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.bx.1 | $328$ | $2$ | $2$ | $1$ |