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: | $8$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}11&64\\22&47\end{bmatrix}$, $\begin{bmatrix}25&64\\67&15\end{bmatrix}$, $\begin{bmatrix}37&16\\68&37\end{bmatrix}$, $\begin{bmatrix}41&32\\62&77\end{bmatrix}$, $\begin{bmatrix}59&64\\19&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.h.1 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $6$ |
| Cyclic 80-torsion field degree: | $192$ |
| Full 80-torsion field degree: | $122880$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 4x $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{720x^{2}y^{14}-391954720x^{2}y^{12}z^{2}+81272125440x^{2}y^{10}z^{4}+362236038144x^{2}y^{8}z^{6}+1117347840000x^{2}y^{6}z^{8}+747205754880x^{2}y^{4}z^{10}+193462272000x^{2}y^{2}z^{12}+17175674880x^{2}z^{14}-179248xy^{14}z+4158420480xy^{12}z^{3}-106072514304xy^{10}z^{5}-549437276160xy^{8}z^{7}-564280950784xy^{6}z^{9}-193226342400xy^{4}z^{11}-21475885056xy^{2}z^{13}-y^{16}+16934400y^{14}z^{2}-24938268928y^{12}z^{4}+138018631680y^{10}z^{6}-6938673152y^{8}z^{8}+105649274880y^{6}z^{10}-7453802496y^{4}z^{12}+754974720y^{2}z^{14}-16777216z^{16}}{y^{2}(x^{2}y^{12}+800x^{2}y^{10}z^{2}-61056x^{2}y^{8}z^{4}+1112064x^{2}y^{6}z^{6}+1290240x^{2}y^{4}z^{8}-100270080x^{2}y^{2}z^{10}+268697600x^{2}z^{12}+24xy^{12}z-1824xy^{10}z^{3}+145408xy^{8}z^{5}-5707776xy^{6}z^{7}+75595776xy^{4}z^{9}-268369920xy^{2}z^{11}+224y^{12}z^{2}-22272y^{10}z^{4}+851712y^{8}z^{6}-12304384y^{6}z^{8}+51314688y^{4}z^{10}+1572864y^{2}z^{12}+1048576z^{14})}$ |
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.g.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-16.g.1.16 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.0-8.q.1.4 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 80.48.1-16.b.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | 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.m.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.m.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.o.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-16.o.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bk.1.9 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bk.2.12 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bo.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.bo.2.10 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-16.cn.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cs.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cs.2.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.ct.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hc.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.he.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.he.2.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hf.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.480.17-80.p.1.23 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 160.192.3-32.i.1.4 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-32.i.2.2 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.i.1.10 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.i.2.4 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-32.j.1.3 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-32.j.2.2 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.j.1.11 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.j.2.7 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.5-32.i.1.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.i.2.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.i.1.13 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.i.2.13 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.j.1.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-32.j.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.j.1.24 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.j.2.24 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-48.bk.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bk.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bo.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bo.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ee.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ee.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.em.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.em.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-48.ga.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gc.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gc.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gd.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sj.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sn.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sn.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sp.1.7 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-48.bf.1.15 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.ml.1.43 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |