Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}21&0\\42&69\end{bmatrix}$, $\begin{bmatrix}23&0\\24&31\end{bmatrix}$, $\begin{bmatrix}33&40\\62&29\end{bmatrix}$, $\begin{bmatrix}69&8\\57&35\end{bmatrix}$, $\begin{bmatrix}71&24\\76&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.g.1 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $12$ |
| Cyclic 80-torsion field degree: | $384$ |
| Full 80-torsion field degree: | $122880$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{12490x^{2}y^{12}z^{2}-15913131x^{2}y^{8}z^{6}+113246025x^{2}y^{4}z^{10}-16777215x^{2}z^{14}-188xy^{14}z+3354639xy^{10}z^{5}-104857784xy^{6}z^{9}+83886081xy^{2}z^{13}+y^{16}-335468y^{12}z^{4}+41954972y^{8}z^{8}-67109046y^{4}z^{12}+z^{16}}{zy^{4}(x^{2}y^{8}z-x^{2}y^{4}z^{5}-x^{2}z^{9}-xy^{10}-xy^{2}z^{8}+2y^{8}z^{3}-2y^{4}z^{7}-z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 80.48.0-16.h.1.1 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-16.h.1.15 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-8.q.1.1 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.1-16.a.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-16.a.1.16 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.192.1-16.l.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.l.2.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.n.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.n.2.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bh.1.10 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bh.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bl.1.11 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bl.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-16.cn.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.co.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cp.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cp.2.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gy.1.13 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gz.1.9 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ha.1.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.ha.2.11 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.17-80.m.1.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 240.192.1-48.bh.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bh.2.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bl.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bl.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dx.1.17 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.dx.2.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ef.1.17 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ef.2.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-48.fw.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fx.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fy.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fy.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sa.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sb.1.25 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sc.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sc.2.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-48.y.1.33 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.mg.1.7 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |