Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8P0 |
| Sutherland and Zywina (SZ) label: | 8P0-8a |
| Rouse and Zureick-Brown (RZB) label: | X196 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.204 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&4\\0&7\end{bmatrix}$, $\begin{bmatrix}5&4\\2&3\end{bmatrix}$, $\begin{bmatrix}5&7\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\wr C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^8}\cdot\frac{(x+y)^{48}(23x^{8}+136x^{7}y+488x^{6}y^{2}-1232x^{5}y^{3}-2392x^{4}y^{4}-10400x^{3}y^{5}-8032x^{2}y^{6}+13888xy^{7}+5744y^{8})^{3}(41x^{8}-8x^{7}y+344x^{6}y^{2}+208x^{5}y^{3}+344x^{4}y^{4}-2912x^{3}y^{5}+5216x^{2}y^{6}+21952xy^{7}+19856y^{8})^{3}}{(x-2y)^{4}(x+y)^{52}(x^{2}+2y^{2})^{8}(x^{2}-4xy-14y^{2})^{8}(x^{2}+2xy+10y^{2})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0.bf.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.e.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.k.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.l.1 | $8$ | $2$ | $2$ | $1$ |
| 8.96.1.n.1 | $8$ | $2$ | $2$ | $1$ |
| 24.96.1.dt.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.du.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dv.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.dw.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.6.d.2 | $24$ | $3$ | $3$ | $6$ |
| 24.192.9.rw.2 | $24$ | $4$ | $4$ | $9$ |
| 40.96.1.cq.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.cr.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.cs.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.ct.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.16.cj.2 | $40$ | $5$ | $5$ | $16$ |
| 40.288.15.ga.2 | $40$ | $6$ | $6$ | $15$ |
| 40.480.31.il.2 | $40$ | $10$ | $10$ | $31$ |
| 56.96.1.co.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.cp.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.cq.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.cr.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.25.ru.1 | $56$ | $8$ | $8$ | $25$ |
| 56.1008.68.d.2 | $56$ | $21$ | $21$ | $68$ |
| 56.1344.93.ij.2 | $56$ | $28$ | $28$ | $93$ |
| 88.96.1.co.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.cp.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.cq.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1.cr.1 | $88$ | $2$ | $2$ | $1$ |
| 104.96.1.cq.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.cr.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.cs.1 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1.ct.1 | $104$ | $2$ | $2$ | $1$ |
| 120.96.1.tx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.tz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ub.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ud.1 | $120$ | $2$ | $2$ | $1$ |
| 136.96.1.cq.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.cr.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.cs.1 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1.ct.1 | $136$ | $2$ | $2$ | $1$ |
| 152.96.1.co.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.cp.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.cq.1 | $152$ | $2$ | $2$ | $1$ |
| 152.96.1.cr.1 | $152$ | $2$ | $2$ | $1$ |
| 168.96.1.tq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ts.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.tu.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.tw.1 | $168$ | $2$ | $2$ | $1$ |
| 184.96.1.co.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.cp.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.cq.1 | $184$ | $2$ | $2$ | $1$ |
| 184.96.1.cr.1 | $184$ | $2$ | $2$ | $1$ |
| 232.96.1.cq.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.cr.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.cs.1 | $232$ | $2$ | $2$ | $1$ |
| 232.96.1.ct.1 | $232$ | $2$ | $2$ | $1$ |
| 248.96.1.co.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.cp.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.cq.1 | $248$ | $2$ | $2$ | $1$ |
| 248.96.1.cr.1 | $248$ | $2$ | $2$ | $1$ |
| 264.96.1.tq.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ts.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.tu.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.tw.1 | $264$ | $2$ | $2$ | $1$ |
| 280.96.1.qj.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.ql.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.qn.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.qp.1 | $280$ | $2$ | $2$ | $1$ |
| 296.96.1.cq.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.cr.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.cs.1 | $296$ | $2$ | $2$ | $1$ |
| 296.96.1.ct.1 | $296$ | $2$ | $2$ | $1$ |
| 312.96.1.tt.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.tv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.tx.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.tz.1 | $312$ | $2$ | $2$ | $1$ |
| 328.96.1.cq.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.cr.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.cs.1 | $328$ | $2$ | $2$ | $1$ |
| 328.96.1.ct.1 | $328$ | $2$ | $2$ | $1$ |