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: | $8$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.123 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&7\\0&13\end{bmatrix}$, $\begin{bmatrix}11&11\\0&5\end{bmatrix}$, $\begin{bmatrix}13&12\\0&13\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.d.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $1$ |
| 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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\,\frac{360x^{2}y^{14}-24497170x^{2}y^{12}z^{2}+634938480x^{2}y^{10}z^{4}+353746131x^{2}y^{8}z^{6}+136395000x^{2}y^{6}z^{8}+11401455x^{2}y^{4}z^{10}+369000x^{2}y^{2}z^{12}+4095x^{2}z^{14}-44812xy^{14}z+129950640xy^{12}z^{3}-414345759xy^{10}z^{5}-268279920xy^{8}z^{7}-34440976xy^{6}z^{9}-1474200xy^{4}z^{11}-20481xy^{2}z^{13}-y^{16}+2116800y^{14}z^{2}-389660452y^{12}z^{4}+269567640y^{10}z^{6}-1694012y^{8}z^{8}+3224160y^{6}z^{10}-28434y^{4}z^{12}+360y^{2}z^{14}-z^{16}}{y^{2}(x^{2}y^{12}+100x^{2}y^{10}z^{2}-954x^{2}y^{8}z^{4}+2172x^{2}y^{6}z^{6}+315x^{2}y^{4}z^{8}-3060x^{2}y^{2}z^{10}+1025x^{2}z^{12}+12xy^{12}z-114xy^{10}z^{3}+1136xy^{8}z^{5}-5574xy^{6}z^{7}+9228xy^{4}z^{9}-4095xy^{2}z^{11}+56y^{12}z^{2}-696y^{10}z^{4}+3327y^{8}z^{6}-6008y^{6}z^{8}+3132y^{4}z^{10}+12y^{2}z^{12}+z^{14})}$ |
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.k.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.g.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.g.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.k.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.11 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.g.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.g.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.h.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.h.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 32.192.3-32.e.1.2 | $32$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 32.192.3-32.f.1.2 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.f.2.4 | $32$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 32.192.3-32.g.1.2 | $32$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.1-48.s.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.s.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.t.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.t.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.j.1.3 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 48.384.9-48.id.1.3 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 80.192.1-80.s.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.s.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.t.1.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.t.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.f.1.9 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.3-96.e.1.3 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.f.1.1 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.f.2.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.g.1.3 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.1-112.s.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.s.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.t.1.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.t.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 160.192.3-160.e.1.3 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.f.1.2 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.f.2.6 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.192.3-160.g.1.3 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.192.1-176.s.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.s.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.t.1.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.t.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.s.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.s.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.t.1.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.t.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 224.192.3-224.e.1.4 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.f.1.2 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.f.2.6 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.192.3-224.g.1.4 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.1-240.cc.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.cc.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.cd.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.cd.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.s.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.s.2.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.t.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.t.2.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.s.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.s.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.t.1.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.t.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |