Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| Elliptic points: | $0$ 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: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.1.116 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&11\\0&15\end{bmatrix}$, $\begin{bmatrix}9&4\\0&9\end{bmatrix}$, $\begin{bmatrix}13&11\\8&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $\OD_{32}:C_2^2$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.s.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{5}$ |
| 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 y^{2} - w^{2} $ |
| $=$ | $2 x^{2} - y^{2} + 2 y w + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 12 x^{2} z^{2} + 4 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 to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^{16}}\cdot\frac{525931019712yz^{22}w+17448204777120yz^{20}w^{3}-1986344744544yz^{18}w^{5}-135039533179248yz^{16}w^{7}+39319410531456yz^{14}w^{9}+234142912377792yz^{12}w^{11}+27060281478720yz^{10}w^{13}-136890066083040yz^{8}w^{15}-89883199990080yz^{6}w^{17}-23935903215840yz^{4}w^{19}-2965510133088yz^{2}w^{21}-141214768240yw^{23}-18391932981z^{24}-3886800750288z^{22}w^{2}-27945863642412z^{20}w^{4}+61060795493400z^{18}w^{6}+130289309349489z^{16}w^{8}-157569205524192z^{14}w^{10}-222792773014872z^{12}w^{12}+51702256914480z^{10}w^{14}+156201185513457z^{8}w^{16}+80045237798640z^{6}w^{18}+18710956791828z^{4}w^{20}+2118221523608z^{2}w^{22}+94143178827w^{24}}{z^{16}(12096yz^{6}w-4896yz^{4}w^{3}-16416yz^{2}w^{5}-3280yw^{7}-3456z^{8}-12528z^{6}w^{2}+14748z^{4}w^{4}+14216z^{2}w^{6}+2187w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.96.1.s.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-2X^{2}Y^{2}-12X^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.p.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.l.2.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.l.2.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.p.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.bb.2.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.bb.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.j.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.j.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.v.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.v.2.10 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.x.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.x.1.8 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.384.5-16.ch.1.4 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.ci.1.3 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.r.1.1 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.bg.1.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.9-32.bq.2.4 | $32$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 32.384.9-32.bs.1.4 | $32$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.384.5-48.ih.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.ii.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.qt.1.12 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.qv.1.12 | $48$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 80.384.5-80.nr.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ns.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bj.2.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.cs.1.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.9-96.ea.2.8 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.ee.1.7 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.5-112.ih.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.ii.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cj.2.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ei.1.12 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.9-160.ee.2.7 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.9-160.ei.1.7 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.5-176.ih.1.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.ii.1.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.nr.1.8 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ns.1.8 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bj.2.7 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.cs.1.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.9-224.ea.2.8 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.9-224.ee.1.7 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-240.cap.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.caq.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.nr.1.8 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.ns.2.6 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.ih.1.8 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.ii.1.8 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |