Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H1 |
| Rouse and Zureick-Brown (RZB) label: | X352 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.116 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&12\\2&7\end{bmatrix}$, $\begin{bmatrix}13&5\\10&3\end{bmatrix}$, $\begin{bmatrix}13&7\\2&3\end{bmatrix}$, $\begin{bmatrix}13&7\\8&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 16-isogeny field degree: | $4$ |
| Cyclic 16-torsion field degree: | $32$ |
| Full 16-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y - x z - y w $ |
| $=$ | $x^{2} - 2 y^{2} + 4 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{3} z + 2 x^{2} y^{2} - x^{2} z^{2} - 8 x y^{2} z + 4 x z^{3} + 4 y^{2} z^{2} - 2 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{21233440xz^{10}w-58386624xz^{8}w^{3}+42403824xz^{6}w^{5}-107606960xz^{4}w^{7}-65128990xz^{2}w^{9}-9175040xw^{11}-102629408y^{2}z^{10}+288426288y^{2}z^{8}w^{2}-338611472y^{2}z^{6}w^{4}+72216568y^{2}z^{4}w^{6}-190950042y^{2}z^{2}w^{8}-27720961y^{2}w^{10}+247726080yz^{11}-729022176yz^{9}w^{2}+846975296yz^{7}w^{4}-568088976yz^{5}w^{6}+25540560yz^{3}w^{8}-71451102yzw^{10}-145096640z^{12}+452980640z^{10}w^{2}-546385504z^{8}w^{4}+482295824z^{6}w^{6}-40228204z^{4}w^{8}-29986366z^{2}w^{10}-12976128w^{12}}{12288xz^{10}w-15360xz^{8}w^{3}+6216xz^{6}w^{5}-856xz^{4}w^{7}+18xz^{2}w^{9}-59392y^{2}z^{10}+80896y^{2}z^{8}w^{2}-37880y^{2}z^{6}w^{4}+6732y^{2}z^{4}w^{6}-314y^{2}z^{2}w^{8}+y^{2}w^{10}+143360yz^{11}-212992yz^{9}w^{2}+112640yz^{7}w^{4}-24664yz^{5}w^{6}+1864yz^{3}w^{8}-22yzw^{10}-83968z^{12}+139264z^{10}w^{2}-84496z^{8}w^{4}+22216z^{6}w^{6}-2220z^{4}w^{8}+38z^{2}w^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bi.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.3.d.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.bu.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.co.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.db.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.ep.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.eq.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.et.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.eu.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.5.cz.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.96.5.da.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.96.5.dd.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 16.96.5.de.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.3.sb.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.sh.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.sv.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.tb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.wj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.wk.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.wr.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ws.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.5.pn.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.po.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.pv.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.pw.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.144.7.pq.1 | $48$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 48.192.11.kk.1 | $48$ | $4$ | $4$ | $11$ | $2$ | $1^{10}$ |
| 80.96.3.vl.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.vr.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.wf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.wl.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bab.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bac.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.baj.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bak.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.qj.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.qk.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.qr.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.qs.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.240.17.ic.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 80.288.17.bbg.1 | $80$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 112.96.3.rb.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.rh.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.rv.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.sb.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.vj.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.vk.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.vr.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.vs.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.ov.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.ow.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.pd.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.pe.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.3.rb.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.rh.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.rv.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.sb.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.vj.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.vk.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.vr.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.vs.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.5.ov.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.ow.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.pd.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.pe.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.3.vl.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.vr.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.wf.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.wl.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bab.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bac.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.baj.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bak.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.5.qj.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.qk.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.qr.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.qs.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.3.fdv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.feh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ffj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ffv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fnj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fnk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fnz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.foa.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.dkl.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dkm.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dlb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dlc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.3.vl.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.vr.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.wf.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.wl.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bab.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bac.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.baj.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bak.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.5.qj.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.qk.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.qr.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.qs.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.3.rb.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.rh.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.rv.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.sb.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.vj.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.vk.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.vr.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.vs.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.5.ov.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.ow.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.pd.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.pe.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |