Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| 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: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.1.49 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&0\\0&13\end{bmatrix}$, $\begin{bmatrix}1&9\\0&3\end{bmatrix}$, $\begin{bmatrix}1&10\\0&1\end{bmatrix}$, $\begin{bmatrix}9&12\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.C_2^3$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.m.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $1$ |
| Cyclic 16-torsion field degree: | $2$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
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 |
|---|
| $(0:0:1)$, $(0:1:0)$, $(-1:0:1)$, $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{184308x^{2}y^{28}z^{2}-5367972750x^{2}y^{24}z^{6}+41764935591x^{2}y^{20}z^{10}-258683339925x^{2}y^{16}z^{14}+80941084824x^{2}y^{12}z^{18}-7279406235x^{2}y^{8}z^{22}+741x^{2}y^{4}z^{26}+16777215x^{2}z^{30}-728xy^{30}z+460224306xy^{26}z^{5}-70213708728xy^{22}z^{9}+34384394057xy^{18}z^{13}-127397006496xy^{14}z^{17}+42058574265xy^{10}z^{21}-4781507300xy^{6}z^{25}+184549377xy^{2}z^{29}+y^{32}-18018552y^{28}z^{4}+32538576668y^{24}z^{8}+123405918974y^{20}z^{12}-163744927716y^{16}z^{16}+44972779664y^{12}z^{20}-4630319234y^{8}z^{24}+167771418y^{4}z^{28}+z^{32}}{zy^{4}(17x^{2}y^{24}z-3754x^{2}y^{20}z^{5}+77190x^{2}y^{16}z^{9}+14156102x^{2}y^{12}z^{13}+61145087x^{2}y^{8}z^{17}+30932985x^{2}y^{4}z^{21}+1048575x^{2}z^{25}+xy^{26}-16xy^{22}z^{4}-60990xy^{18}z^{8}+5373660xy^{14}z^{12}+57737227xy^{10}z^{16}+69730312xy^{6}z^{20}+9437185xy^{2}z^{24}+92y^{24}z^{3}-21164y^{20}z^{7}+1110895y^{16}z^{11}+25427620y^{12}z^{15}+46137336y^{8}z^{19}+8388614y^{4}z^{23}+z^{27})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.n.2.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.n.2.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.u.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.u.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.v.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.v.1.10 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.h.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.h.1.10 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.u.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.u.1.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.v.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.v.1.11 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.384.5-16.bu.2.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.bw.1.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.bx.2.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.bx.4.3 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.by.3.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.by.4.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.n.2.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.p.2.3 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.t.1.6 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.u.2.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.v.3.3 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.384.5-32.v.4.3 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.384.5-32.bc.2.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.be.2.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.9-32.bh.1.2 | $32$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 32.384.9-32.bh.2.3 | $32$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.5-48.gr.2.3 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.gs.2.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.gt.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gt.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gu.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gu.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.md.1.2 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.ov.1.3 | $48$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 80.384.5-80.lp.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.lq.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.lr.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.lr.4.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ls.3.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ls.4.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bf.2.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bh.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bp.1.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bq.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.br.3.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.br.4.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.co.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.cq.2.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.9-96.cx.1.2 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.cx.2.3 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.5-112.gr.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gs.2.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gt.2.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gt.4.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gu.3.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gu.4.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cf.2.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ch.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ct.1.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cu.2.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cv.3.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cv.4.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ee.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.eg.2.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.9-160.db.1.1 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 160.384.9-160.db.2.1 | $160$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.5-176.gr.2.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gs.2.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gt.2.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gt.4.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gu.3.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gu.4.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lp.2.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lq.2.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lr.2.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lr.4.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ls.3.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ls.4.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bf.2.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bh.2.6 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bp.1.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bq.2.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.br.3.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.br.4.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.co.2.6 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.cq.2.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.9-224.cx.1.2 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 224.384.9-224.cx.2.2 | $224$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-240.bri.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brk.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brl.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brl.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brm.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brm.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lp.2.3 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lq.2.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lr.2.5 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lr.4.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.ls.2.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.ls.4.5 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gr.2.3 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gs.2.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gt.2.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gt.4.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gu.3.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gu.4.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |