Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 | $2^{8}$ | ||
| 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: | 16.192.1.57 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}9&8\\8&13\end{bmatrix}$, $\begin{bmatrix}9&14\\8&5\end{bmatrix}$, $\begin{bmatrix}11&11\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.D_4$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.l.1 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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y^{2} - 2 y z - z^{2} $ |
| $=$ | $4 y z - 10 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\,\frac{1073741824y^{24}+12884901888y^{22}w^{2}+65229815808y^{20}w^{4}+181059715072y^{18}w^{6}+299712380928y^{16}w^{8}+301266370560y^{14}w^{10}+178963349504y^{12}w^{12}+57901645824y^{10}w^{14}+8402565120y^{8}w^{16}+262407680y^{6}w^{18}-13657536y^{4}w^{20}+188592y^{2}w^{22}-3814697265624999936z^{24}+1831054687499999232z^{22}w^{2}-347900390624996352z^{20}w^{4}+32470703124991488z^{18}w^{6}-1417236328115712z^{16}w^{8}+10253906248320z^{14}w^{10}+1385742181824z^{12}w^{12}-42539058144z^{10}w^{14}-142089480z^{8}w^{16}+31716720z^{6}w^{18}-571896z^{4}w^{20}-7092z^{2}w^{22}-2213w^{24}}{w^{16}(128y^{8}+512y^{6}w^{2}+544y^{4}w^{4}+112y^{2}w^{6}-195312z^{8}+31248z^{6}w^{2}-936z^{4}w^{4}-12z^{2}w^{6}-3w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.96.1.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+X^{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.n.2.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.n.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.g.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.g.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.bu.2.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.bv.1.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.384.5-32.o.1.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.s.2.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.w.2.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.bd.1.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gp.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.gq.1.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.576.17-48.lw.2.3 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.oq.2.2 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 80.384.5-80.ln.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.lo.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bg.1.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bo.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bs.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.cp.1.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gp.2.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gq.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cg.1.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cs.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cw.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ef.1.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gp.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gq.1.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ln.1.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lo.1.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bg.1.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bo.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bs.1.5 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.cp.1.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brd.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bre.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.ln.2.2 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lo.1.2 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gp.1.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gq.1.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |