Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ 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: | 8G0 |
| Rouse and Zureick-Brown (RZB) label: | X75d |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.0.206 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&8\\8&1\end{bmatrix}$, $\begin{bmatrix}7&14\\8&15\end{bmatrix}$, $\begin{bmatrix}11&15\\8&13\end{bmatrix}$, $\begin{bmatrix}11&15\\8&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.k.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2}\cdot\frac{x^{24}(x^{4}-16x^{3}y-112x^{2}y^{2}-128xy^{3}+64y^{4})^{3}(x^{4}+16x^{3}y-112x^{2}y^{2}+128xy^{3}+64y^{4})^{3}}{y^{2}x^{26}(x^{2}-8y^{2})^{2}(x^{2}+8y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 16.96.0-8.m.1.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-8.m.1.4 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-8.m.2.2 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-8.m.2.4 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.e.1.3 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.e.1.7 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.f.1.5 | $16$ | $2$ | $2$ | $0$ |
| 16.96.0-16.f.1.6 | $16$ | $2$ | $2$ | $0$ |
| 16.96.1-16.d.1.2 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.d.1.4 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.f.1.5 | $16$ | $2$ | $2$ | $1$ |
| 16.96.1-16.f.1.7 | $16$ | $2$ | $2$ | $1$ |
| 48.96.0-24.bd.1.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bd.1.6 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bd.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-24.bd.2.4 | $48$ | $2$ | $2$ | $0$ |
| 48.144.4-24.cp.1.13 | $48$ | $3$ | $3$ | $4$ |
| 48.192.3-24.cr.1.13 | $48$ | $4$ | $4$ | $3$ |
| 80.96.0-40.be.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-40.be.1.7 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-40.be.2.5 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-40.be.2.7 | $80$ | $2$ | $2$ | $0$ |
| 80.240.8-40.z.1.7 | $80$ | $5$ | $5$ | $8$ |
| 80.288.7-40.bx.1.16 | $80$ | $6$ | $6$ | $7$ |
| 80.480.15-40.cp.1.14 | $80$ | $10$ | $10$ | $15$ |
| 48.96.0-48.e.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.e.1.10 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.f.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.f.1.11 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-48.d.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.d.1.6 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.f.1.2 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.f.1.6 | $48$ | $2$ | $2$ | $1$ |
| 112.96.0-56.bc.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-56.bc.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.384.11-56.bt.1.14 | $112$ | $8$ | $8$ | $11$ |
| 80.96.0-80.g.1.9 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.g.1.11 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.h.1.9 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.h.1.10 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.d.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.d.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.f.1.1 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.f.1.9 | $80$ | $2$ | $2$ | $1$ |
| 176.96.0-88.bc.1.1 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-88.bc.1.5 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-88.bc.2.1 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-88.bc.2.3 | $176$ | $2$ | $2$ | $0$ |
| 208.96.0-104.be.1.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.be.1.5 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.be.2.1 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-104.be.2.3 | $208$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.10 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.9 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.11 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.d.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.d.1.10 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.f.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.f.1.10 | $112$ | $2$ | $2$ | $1$ |
| 240.96.0-120.db.1.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.db.1.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.db.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-120.db.2.5 | $240$ | $2$ | $2$ | $0$ |
| 272.96.0-136.be.1.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.be.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.be.2.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.be.2.4 | $272$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bc.1.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bc.1.5 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bc.2.1 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-152.bc.2.3 | $304$ | $2$ | $2$ | $0$ |
| 176.96.0-176.e.1.9 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.e.1.10 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.f.1.9 | $176$ | $2$ | $2$ | $0$ |
| 176.96.0-176.f.1.11 | $176$ | $2$ | $2$ | $0$ |
| 176.96.1-176.d.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.d.1.10 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.f.1.2 | $176$ | $2$ | $2$ | $1$ |
| 176.96.1-176.f.1.10 | $176$ | $2$ | $2$ | $1$ |
| 208.96.0-208.g.1.9 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.g.1.11 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.h.1.9 | $208$ | $2$ | $2$ | $0$ |
| 208.96.0-208.h.1.10 | $208$ | $2$ | $2$ | $0$ |
| 208.96.1-208.d.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.d.1.9 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.f.1.1 | $208$ | $2$ | $2$ | $1$ |
| 208.96.1-208.f.1.9 | $208$ | $2$ | $2$ | $1$ |
| 240.96.0-240.g.1.17 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.g.1.19 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.h.1.17 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.h.1.21 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.d.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.d.1.18 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.f.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.f.1.10 | $240$ | $2$ | $2$ | $1$ |
| 272.96.0-272.g.1.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.g.1.11 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.h.1.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.h.1.11 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.d.1.9 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.d.1.15 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.f.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.f.1.10 | $272$ | $2$ | $2$ | $1$ |
| 304.96.0-304.e.1.9 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.e.1.10 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.f.1.9 | $304$ | $2$ | $2$ | $0$ |
| 304.96.0-304.f.1.11 | $304$ | $2$ | $2$ | $0$ |
| 304.96.1-304.d.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.d.1.10 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.f.1.2 | $304$ | $2$ | $2$ | $1$ |
| 304.96.1-304.f.1.10 | $304$ | $2$ | $2$ | $1$ |