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 $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.132 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&3\\0&1\end{bmatrix}$, $\begin{bmatrix}11&5\\8&5\end{bmatrix}$, $\begin{bmatrix}11&11\\0&9\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.D_8$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.r.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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 44x + 112 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(4:0:1)$, $(6:8:1)$, $(6:-8:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^8}\cdot\frac{48x^{2}y^{14}-345376x^{2}y^{12}z^{2}+440527872x^{2}y^{10}z^{4}-197883503616x^{2}y^{8}z^{6}+36045077348352x^{2}y^{6}z^{8}-2522513481990144x^{2}y^{4}z^{10}+74853239784210432x^{2}y^{2}z^{12}-800773957736202240x^{2}z^{14}-1264xy^{14}z+4687104xy^{12}z^{3}-4566200064xy^{10}z^{5}+1764897017856xy^{8}z^{7}-294387861028864xy^{6}z^{9}+19874075322089472xy^{4}z^{11}-579212232933507072xy^{2}z^{13}+6131409481126379520xz^{15}-y^{16}+22656y^{14}z^{2}-43268352y^{12}z^{4}+26437152768y^{10}z^{6}-6852870520832y^{8}z^{8}+813138762006528y^{6}z^{10}-45512793468174336y^{4}z^{12}+1187892640327139328y^{2}z^{14}-11713254600743059456z^{16}}{z^{5}y^{2}(1228x^{2}y^{6}z-3600896x^{2}y^{4}z^{3}+2216833024x^{2}y^{2}z^{5}-359689093120x^{2}z^{7}+xy^{8}-22832xy^{6}z^{2}+40778752xy^{4}z^{4}-19701317632xy^{2}z^{6}+2754086961152xz^{8}-48y^{8}z+296448y^{6}z^{3}-275603456y^{4}z^{5}+74192388096y^{2}z^{7}-5261322354688z^{9})}$ |
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.bb.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.2.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.2.13 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.bb.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.11 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.a.1.12 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.c.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.g.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.l.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.q.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cg.2.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ck.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cw.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.da.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ej.1.14 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bas.1.13 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cf.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cj.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cv.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cz.1.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.bx.1.6 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.cf.2.8 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cj.2.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cv.1.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cz.1.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cf.2.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cj.2.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cv.1.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cz.1.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cf.1.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cj.1.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cv.1.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cz.1.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.iw.2.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.je.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kc.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kk.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cf.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cj.1.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cv.2.4 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cz.2.8 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cf.2.8 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cj.2.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cv.1.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cz.1.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |