Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $8\cdot16$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16B0 |
| Sutherland and Zywina (SZ) label: | 16B0-16d |
| Rouse and Zureick-Brown (RZB) label: | X104 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.0.22 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&14\\8&3\end{bmatrix}$, $\begin{bmatrix}11&1\\6&5\end{bmatrix}$, $\begin{bmatrix}11&5\\6&9\end{bmatrix}$, $\begin{bmatrix}15&9\\14&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $64$ |
| Full 16-torsion field degree: | $1024$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{x^{24}(x^{8}+y^{8})^{3}}{y^{8}x^{40}}$ |
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 |
|---|---|---|---|---|---|
| 8.12.0.w.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 16.48.1.ci.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cj.1 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.ck.1 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cl.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cu.1 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cv.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cw.2 | $16$ | $2$ | $2$ | $1$ |
| 16.48.1.cx.1 | $16$ | $2$ | $2$ | $1$ |
| 16.48.3.e.1 | $16$ | $2$ | $2$ | $3$ |
| 16.48.3.i.1 | $16$ | $2$ | $2$ | $3$ |
| 16.48.3.t.1 | $16$ | $2$ | $2$ | $3$ |
| 16.48.3.w.2 | $16$ | $2$ | $2$ | $3$ |
| 48.48.1.im.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.in.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.io.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.ip.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.iq.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.ir.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.is.2 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1.it.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.3.fz.2 | $48$ | $2$ | $2$ | $3$ |
| 48.48.3.ga.1 | $48$ | $2$ | $2$ | $3$ |
| 48.48.3.gb.2 | $48$ | $2$ | $2$ | $3$ |
| 48.48.3.gc.2 | $48$ | $2$ | $2$ | $3$ |
| 48.72.0.c.2 | $48$ | $3$ | $3$ | $0$ |
| 48.96.7.cq.1 | $48$ | $4$ | $4$ | $7$ |
| 80.48.1.io.1 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.ip.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.iq.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.ir.1 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.is.1 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.it.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.iu.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1.iv.1 | $80$ | $2$ | $2$ | $1$ |
| 80.48.3.gt.1 | $80$ | $2$ | $2$ | $3$ |
| 80.48.3.gu.1 | $80$ | $2$ | $2$ | $3$ |
| 80.48.3.gv.1 | $80$ | $2$ | $2$ | $3$ |
| 80.48.3.gw.2 | $80$ | $2$ | $2$ | $3$ |
| 80.120.8.bm.2 | $80$ | $5$ | $5$ | $8$ |
| 80.144.7.de.1 | $80$ | $6$ | $6$ | $7$ |
| 80.240.15.di.1 | $80$ | $10$ | $10$ | $15$ |
| 112.48.1.im.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.in.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.io.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.ip.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.iq.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.ir.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.is.2 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.it.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.3.fz.2 | $112$ | $2$ | $2$ | $3$ |
| 112.48.3.ga.1 | $112$ | $2$ | $2$ | $3$ |
| 112.48.3.gb.1 | $112$ | $2$ | $2$ | $3$ |
| 112.48.3.gc.2 | $112$ | $2$ | $2$ | $3$ |
| 112.192.15.bg.2 | $112$ | $8$ | $8$ | $15$ |
| 176.48.1.im.1 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.in.2 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.io.2 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.ip.1 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.iq.1 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.ir.2 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.is.2 | $176$ | $2$ | $2$ | $1$ |
| 176.48.1.it.1 | $176$ | $2$ | $2$ | $1$ |
| 176.48.3.fz.2 | $176$ | $2$ | $2$ | $3$ |
| 176.48.3.ga.1 | $176$ | $2$ | $2$ | $3$ |
| 176.48.3.gb.1 | $176$ | $2$ | $2$ | $3$ |
| 176.48.3.gc.2 | $176$ | $2$ | $2$ | $3$ |
| 176.288.23.bg.2 | $176$ | $12$ | $12$ | $23$ |
| 208.48.1.io.1 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.ip.2 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.iq.2 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.ir.1 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.is.1 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.it.2 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.iu.2 | $208$ | $2$ | $2$ | $1$ |
| 208.48.1.iv.1 | $208$ | $2$ | $2$ | $1$ |
| 208.48.3.gl.1 | $208$ | $2$ | $2$ | $3$ |
| 208.48.3.gm.1 | $208$ | $2$ | $2$ | $3$ |
| 208.48.3.gn.1 | $208$ | $2$ | $2$ | $3$ |
| 208.48.3.go.2 | $208$ | $2$ | $2$ | $3$ |
| 208.336.23.ek.1 | $208$ | $14$ | $14$ | $23$ |
| 240.48.1.bba.1 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbb.2 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbc.2 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbd.1 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbe.1 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbf.2 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbg.2 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1.bbh.1 | $240$ | $2$ | $2$ | $1$ |
| 240.48.3.st.2 | $240$ | $2$ | $2$ | $3$ |
| 240.48.3.su.1 | $240$ | $2$ | $2$ | $3$ |
| 240.48.3.sv.2 | $240$ | $2$ | $2$ | $3$ |
| 240.48.3.sw.2 | $240$ | $2$ | $2$ | $3$ |
| 272.48.1.io.1 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.ip.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.iq.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.ir.1 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.is.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.it.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.iu.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.1.iv.2 | $272$ | $2$ | $2$ | $1$ |
| 272.48.3.gl.2 | $272$ | $2$ | $2$ | $3$ |
| 272.48.3.gm.2 | $272$ | $2$ | $2$ | $3$ |
| 272.48.3.gn.2 | $272$ | $2$ | $2$ | $3$ |
| 272.48.3.go.2 | $272$ | $2$ | $2$ | $3$ |
| 304.48.1.im.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.in.2 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.io.2 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.ip.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.iq.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.ir.2 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.is.2 | $304$ | $2$ | $2$ | $1$ |
| 304.48.1.it.1 | $304$ | $2$ | $2$ | $1$ |
| 304.48.3.fz.2 | $304$ | $2$ | $2$ | $3$ |
| 304.48.3.ga.1 | $304$ | $2$ | $2$ | $3$ |
| 304.48.3.gb.1 | $304$ | $2$ | $2$ | $3$ |
| 304.48.3.gc.2 | $304$ | $2$ | $2$ | $3$ |