Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-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 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}1&40\\28&11\end{bmatrix}$, $\begin{bmatrix}1&48\\52&17\end{bmatrix}$, $\begin{bmatrix}17&68\\8&55\end{bmatrix}$, $\begin{bmatrix}39&16\\24&5\end{bmatrix}$, $\begin{bmatrix}39&68\\60&69\end{bmatrix}$, $\begin{bmatrix}73&24\\16&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.c.1 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $24$ |
| Cyclic 80-torsion field degree: | $384$ |
| Full 80-torsion field degree: | $122880$ |
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
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 80.192.1-8.f.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.f.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.f.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.f.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.g.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.g.1.4 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.g.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-8.g.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.w.1.4 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.w.1.6 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.w.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.w.2.8 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.x.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.x.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.x.2.4 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-40.x.2.8 | $80$ | $2$ | $2$ | $1$ |
| 80.192.2-16.a.1.2 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.a.1.4 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.a.1.6 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.a.1.7 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.b.1.3 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.b.1.8 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.b.1.11 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.b.1.14 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.c.1.1 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.c.1.4 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.c.1.5 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.c.1.16 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.d.1.1 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.d.1.2 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.d.1.3 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-16.d.1.8 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.e.1.4 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.e.1.13 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.e.1.29 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.e.1.32 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.f.1.3 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.f.1.14 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.f.1.24 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.f.1.30 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.g.1.4 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.g.1.13 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.g.1.24 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.g.1.30 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.h.1.5 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.h.1.12 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.h.1.23 | $80$ | $2$ | $2$ | $2$ |
| 80.192.2-80.h.1.32 | $80$ | $2$ | $2$ | $2$ |
| 80.192.3-8.i.1.1 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-8.i.1.2 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-8.j.1.1 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-8.j.1.2 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-40.be.1.1 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-40.be.1.2 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-40.bf.1.4 | $80$ | $2$ | $2$ | $3$ |
| 80.192.3-40.bf.1.6 | $80$ | $2$ | $2$ | $3$ |
| 80.480.16-40.f.1.2 | $80$ | $5$ | $5$ | $16$ |
| 240.192.1-24.w.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.w.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.w.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.w.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.x.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.x.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.x.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-24.x.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cy.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cy.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cy.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cy.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cz.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cz.1.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cz.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-120.cz.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.2-48.a.1.1 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.a.1.16 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.a.1.18 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.a.1.28 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.b.1.1 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.b.1.16 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.b.1.18 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.b.1.24 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.c.1.1 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.c.1.8 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.c.1.12 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.c.1.26 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.d.1.1 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.d.1.8 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.d.1.14 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-48.d.1.26 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.e.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.e.1.31 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.e.1.35 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.e.1.43 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.f.1.2 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.f.1.31 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.f.1.35 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.f.1.39 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.g.1.15 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.g.1.18 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.g.1.49 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.g.1.53 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.h.1.15 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.h.1.18 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.h.1.49 | $240$ | $2$ | $2$ | $2$ |
| 240.192.2-240.h.1.51 | $240$ | $2$ | $2$ | $2$ |
| 240.192.3-24.z.1.1 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-24.z.1.8 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-24.ba.1.1 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-24.ba.1.7 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-120.cw.1.7 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-120.cw.1.8 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-120.cx.1.5 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-120.cx.1.6 | $240$ | $2$ | $2$ | $3$ |
| 240.288.8-24.l.1.1 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-24.i.1.1 | $240$ | $4$ | $4$ | $7$ |