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: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.87 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&14\\8&9\end{bmatrix}$, $\begin{bmatrix}13&5\\0&9\end{bmatrix}$, $\begin{bmatrix}13&6\\8&15\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.t.2 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 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 |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{1488x^{2}y^{14}+13787896096x^{2}y^{12}z^{2}+505152166812672x^{2}y^{10}z^{4}+2461749990779790336x^{2}y^{8}z^{6}+3688871433326264451072x^{2}y^{6}z^{8}+2245667712432631893786624x^{2}y^{4}z^{10}+587918910491320028938371072x^{2}y^{2}z^{12}+55028767359159366680523571200x^{2}z^{14}+787504xy^{14}z+641027529984xy^{12}z^{3}+10405160498298624xy^{10}z^{5}+34334754979127709696xy^{8}z^{7}+40659145953670718685184xy^{6}z^{9}+21101593716190261099364352xy^{4}z^{11}+4918866218257187405023936512xy^{2}z^{13}+421347251198306990005136916480xz^{15}+y^{16}+171854976y^{14}z^{2}+20341907757312y^{12}z^{4}+163695851198435328y^{10}z^{6}+337812738279880835072y^{8}z^{8}+269508227949509615812608y^{6}z^{10}+95840436993211162006388736y^{4}z^{12}+14989484223909615018304339968y^{2}z^{14}+804928727046678093132187303936z^{16}}{y^{2}(x^{2}y^{12}+32x^{2}y^{10}z^{2}-896x^{2}y^{8}z^{4}-8192x^{2}y^{6}z^{6}+700416x^{2}y^{4}z^{8}-917504x^{2}y^{2}z^{10}+262144x^{2}z^{12}+224xy^{10}z^{3}+12800xy^{8}z^{5}+108544xy^{6}z^{7}-2981888xy^{4}z^{9}+3735552xy^{2}z^{11}-1048576xz^{13}-64y^{12}z^{2}-3584y^{10}z^{4}-45312y^{8}z^{6}-409600y^{6}z^{8}-18153472y^{4}z^{10}+25165824y^{2}z^{12}-7340032z^{14})}$ |
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.bb.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.bb.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.13 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.a.1.15 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.1.10 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.h.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.n.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.q.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ci.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cm.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cy.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dc.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ex.1.30 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bay.2.26 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.ch.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cl.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cx.2.7 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.db.1.4 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.cd.2.12 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.ch.1.8 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cl.1.6 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cx.1.8 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.db.1.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ch.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cl.1.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cx.1.8 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.db.1.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ch.2.7 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cl.1.4 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cx.2.7 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.db.1.4 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jc.1.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jk.1.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ki.1.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kq.1.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ch.2.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cl.1.8 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cx.2.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.db.1.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ch.1.8 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cl.1.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cx.1.8 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.db.1.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |