Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}4&67\\47&48\end{bmatrix}$, $\begin{bmatrix}54&11\\63&74\end{bmatrix}$, $\begin{bmatrix}69&18\\16&59\end{bmatrix}$, $\begin{bmatrix}72&35\\59&48\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.w.2 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: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} + 2 x z + y^{2} $ |
| $=$ | $32 x^{2} - 30 x z - 5 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + x^{2} y^{2} + 3 x^{2} z^{2} + 2 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4\cdot7^2}\cdot\frac{907067769487360xz^{11}-2886681150947328xz^{9}w^{2}-931213649608704xz^{7}w^{4}+6319843755595776xz^{5}w^{6}+182112132867840xz^{3}w^{8}+64523764187568xzw^{10}-129574475923456z^{12}+298094008926208z^{10}w^{2}+461330347290624z^{8}w^{4}-1332552037533696z^{6}w^{6}+763060519376064z^{4}w^{8}+48251436231600z^{2}w^{10}-424023618123w^{12}}{w^{2}(1679616xz^{9}-1116857728xz^{7}w^{2}+1326273984xz^{5}w^{4}-280842912xz^{3}w^{6}+8297856xzw^{8}-559872z^{10}+157276624z^{8}w^{2}-56963872z^{6}w^{4}-51460584z^{4}w^{6}+8075592z^{2}w^{8}-64827w^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.1.w.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{8}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+X^{2}Y^{2}+3X^{2}Z^{2}+2Z^{4} $ |
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.ba.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 80.48.0-16.f.2.3 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-16.f.2.10 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-8.ba.2.5 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.1-16.b.1.10 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.48.1-16.b.1.15 | $80$ | $2$ | $2$ | $1$ | $?$ | 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.d.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.j.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.o.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.r.2.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.co.1.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cs.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.de.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.di.2.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.ck.2.13 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 160.192.5-32.t.2.2 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.x.2.2 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.bb.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.bf.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bv.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bz.1.9 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.ct.1.16 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cx.2.15 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-48.cp.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.ct.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.df.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dj.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jr.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jz.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kx.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.lf.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.288.9-48.fm.2.6 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.bbf.2.14 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |