Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 | $1^{4}\cdot2^{2}\cdot4^{2}$ | ||
| 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: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.1.53 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}9&6\\0&9\end{bmatrix}$, $\begin{bmatrix}13&7\\8&11\end{bmatrix}$, $\begin{bmatrix}13&12\\8&1\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.D_4$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.l.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 4x $ |
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 |
|---|
| $(-2:0:1)$, $(2:0:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{2948928x^{2}y^{28}z^{2}-5496804096000x^{2}y^{24}z^{6}+2737106818891776x^{2}y^{20}z^{10}-1084996567380787200x^{2}y^{16}z^{14}+21727457013865119744x^{2}y^{12}z^{18}-125059246854493962240x^{2}y^{8}z^{22}+814738116182016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-2912xy^{30}z+117817422336xy^{26}z^{5}-1150381403799552xy^{22}z^{9}+36054650382712832xy^{18}z^{13}-8549468382947180544xy^{14}z^{17}+180640200984562237440xy^{10}z^{21}-1314330718661456691200xy^{6}z^{25}+3246626974565067128832xy^{2}z^{29}+y^{32}-1153187328y^{28}z^{4}+133278010032128y^{24}z^{8}+32350121223520256y^{20}z^{12}-2747184021195718656y^{16}z^{16}+48289154466773467136y^{12}z^{20}-318193114881116340224y^{8}z^{24}+737866499597870825472y^{4}z^{28}+281474976710656z^{32}}{zy^{4}(68x^{2}y^{24}z-961024x^{2}y^{20}z^{5}+1264680960x^{2}y^{16}z^{9}+14843748810752x^{2}y^{12}z^{13}+4103377327751168x^{2}y^{8}z^{17}+132856158942658560x^{2}y^{4}z^{21}+288230101273804800x^{2}z^{25}+xy^{26}-1024xy^{22}z^{4}-249815040xy^{18}z^{8}+1408672727040xy^{14}z^{12}+968669928620032xy^{10}z^{16}+74872352394969088xy^{6}z^{20}+648518415060828160xy^{2}z^{24}+1472y^{24}z^{3}-21671936y^{20}z^{7}+72803614720y^{16}z^{11}+106651168276480y^{12}z^{15}+12384896827785216y^{8}z^{19}+144115291155070976y^{4}z^{23}+1099511627776z^{27})}$ |
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.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-8.n.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.j.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.2.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.y.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.2.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.g.1.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.g.1.12 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.2.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.q.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.2.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.96.1-16.r.2.5 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.384.5-16.bu.1.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.bv.1.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 32.384.5-32.o.2.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.s.1.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.w.1.4 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 32.384.5-32.bd.2.2 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gp.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.gq.2.3 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.576.17-48.lw.1.2 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.oq.1.4 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 80.384.5-80.ln.2.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.lo.2.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bg.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bo.2.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bs.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.cp.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gp.1.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.gq.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cg.2.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cs.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.cw.2.3 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.384.5-160.ef.2.4 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gp.1.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.gq.2.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ln.2.3 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.lo.2.3 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bg.2.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bo.2.3 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.bs.2.2 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.384.5-224.cp.2.4 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brd.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bre.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.ln.1.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.lo.2.3 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gp.2.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.384.5-304.gq.2.3 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |