Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| 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: | 16.96.1.109 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}3&5\\8&15\end{bmatrix}$, $\begin{bmatrix}7&15\\0&5\end{bmatrix}$, $\begin{bmatrix}11&7\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $D_8:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.u.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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x - 14 $ |
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 |
|---|
| $(0:1:0)$, $(-2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{24x^{2}y^{14}-23054x^{2}y^{12}z^{2}+12222744x^{2}y^{10}z^{4}-116093168151x^{2}y^{8}z^{6}-4373456358144x^{2}y^{6}z^{8}+83710621777209x^{2}y^{4}z^{10}-451113652322316x^{2}y^{2}z^{12}+782005818097665x^{2}z^{14}-404xy^{14}z+2313672xy^{12}z^{3}-1530994881xy^{10}z^{5}-606689937078xy^{8}z^{7}-14055316122824xy^{6}z^{9}+306547104155016xy^{4}z^{11}-1703337439264743xy^{2}z^{13}+2993852285714430xz^{15}+y^{16}-14448y^{14}z^{2}+24081828y^{12}z^{4}-17813419656y^{10}z^{6}-1580695580788y^{8}z^{8}+4159445162016y^{6}z^{10}+200154653135754y^{4}z^{12}-1468049282564016y^{2}z^{14}+2859681299038201z^{16}}{zy^{2}(335x^{2}y^{10}z+202928x^{2}y^{8}z^{3}+33767533x^{2}y^{6}z^{5}+2280914948x^{2}y^{4}z^{7}+67234103297x^{2}y^{2}z^{9}+719378186240x^{2}z^{11}+xy^{12}+3542xy^{10}z^{2}+1372012xy^{8}z^{4}+181174064xy^{6}z^{6}+10541006841xy^{4}z^{8}+279219666942xy^{2}z^{10}+2754086961152xz^{12}+24y^{12}z+27734y^{10}z^{3}+6422544y^{8}z^{5}+568720606y^{6}z^{7}+22851616736y^{4}z^{9}+412928704505y^{2}z^{11}+2630661177344z^{13})}$ |
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.ba.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.ba.1.7 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.b.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | 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.d.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.i.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.m.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.r.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 32.192.5-32.r.1.8 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.v.1.8 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 32.192.5-32.z.1.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.192.5-32.bd.1.2 | $32$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.1-48.cn.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cr.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dd.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dh.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ey.2.3 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.baz.1.4 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.cm.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cq.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.dc.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.dg.2.3 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.ce.2.3 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.5-96.bt.1.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bx.1.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cr.2.15 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cv.2.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.1-112.cm.1.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cq.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.dc.2.5 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.dg.2.3 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 160.192.5-160.bt.1.14 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.bx.2.10 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cr.2.8 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.192.5-160.cv.2.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.1-176.cm.1.5 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cq.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.dc.2.3 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.dg.2.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cm.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cq.2.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.dc.2.3 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.dg.2.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 224.192.5-224.bt.1.15 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.bx.2.13 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cr.2.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.192.5-224.cv.2.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.jl.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jt.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kr.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kz.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cm.2.3 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cq.2.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.dc.1.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.dg.1.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cm.1.5 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cq.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.dc.2.3 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.dg.2.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |