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: | 16.96.1.254 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&4\\8&9\end{bmatrix}$, $\begin{bmatrix}5&0\\0&11\end{bmatrix}$, $\begin{bmatrix}13&5\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.Q_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.s.1 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 $ | $=$ | $ 2 x^{2} + 2 x z + y^{2} $ |
| $=$ | $17 x^{2} - 14 x z - 2 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4\cdot7^2}\cdot\frac{14172933898240xz^{11}-90208785967104xz^{9}w^{2}-58200853100544xz^{7}w^{4}+789980469449472xz^{5}w^{6}+45528033216960xz^{3}w^{8}+32261882093784xzw^{10}-2024601186304z^{12}+9315437778944z^{10}w^{2}+28833146705664z^{8}w^{4}-166569004691712z^{6}w^{6}+190765129844016z^{4}w^{8}+24125718115800z^{2}w^{10}-424023618123w^{12}}{w^{2}(52488xz^{9}-69803608xz^{7}w^{2}+165784248xz^{5}w^{4}-70210728xz^{3}w^{6}+4148928xzw^{8}-17496z^{10}+9829789z^{8}w^{2}-7120484z^{6}w^{4}-12865146z^{4}w^{6}+4037796z^{2}w^{8}-64827w^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.48.1.s.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{8}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{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 |
|---|---|---|---|---|---|---|
| 8.48.0-8.ba.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.2.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.ba.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.9 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.a.1.14 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.h.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.n.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.p.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ch.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cl.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cx.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.db.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ew.2.1 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bax.2.3 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cg.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.ck.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cw.2.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.da.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.cc.2.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.cg.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.ck.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cw.2.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.da.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cg.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ck.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cw.2.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.da.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cg.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ck.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cw.2.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.da.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jb.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jj.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kh.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kp.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cg.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ck.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cw.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.da.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cg.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ck.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cw.2.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.da.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |