Invariants
| Level: | $16$ | $\SL_2$-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: | $8$ (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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.83 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&6\\8&9\end{bmatrix}$, $\begin{bmatrix}5&10\\0&11\end{bmatrix}$, $\begin{bmatrix}13&3\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.Q_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.s.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $256$ |
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^{2} + x z + y^{2} $ |
| $=$ | $35 x^{2} - 27 x z - 7 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 2 x^{2} y^{2} + 3 x^{2} z^{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.s.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{8}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}y$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+2X^{2}Y^{2}+3X^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.ba.2.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.ba.2.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.8 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.a.1.15 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.192.1-16.a.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.h.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.n.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.p.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ch.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cl.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cx.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.db.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ew.1.30 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bax.1.29 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cg.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.ck.2.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cw.1.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.da.2.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.cc.1.14 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.cg.2.6 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.ck.2.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cw.1.8 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.da.2.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cg.2.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ck.2.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cw.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.da.2.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cg.2.7 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ck.2.4 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cw.1.7 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.da.2.4 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jb.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jj.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kh.1.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kp.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cg.1.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ck.1.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cw.2.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.da.2.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cg.2.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ck.2.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cw.1.8 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.da.2.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |