Invariants
Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $256$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $2\cdot4$ | ||
Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16I1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.8 |
Level structure
$\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}5&1\\10&3\end{bmatrix}$, $\begin{bmatrix}5&15\\0&3\end{bmatrix}$, $\begin{bmatrix}11&3\\10&5\end{bmatrix}$, $\begin{bmatrix}11&8\\4&15\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 16-isogeny field degree: | $8$ |
Cyclic 16-torsion field degree: | $64$ |
Full 16-torsion field degree: | $512$ |
Jacobian
Conductor: | $2^{8}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 256.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y z + z^{2} + w^{2} $ |
$=$ | $2 x^{2} + y^{2} - y z + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 2 x^{2} y^{2} + 4 x^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{27y^{12}+540y^{10}w^{2}+4194y^{8}w^{4}+16784y^{6}w^{6}+44355y^{4}w^{8}+113556y^{2}w^{10}+4069z^{12}+48288z^{10}w^{2}+246672z^{8}w^{4}+697664z^{6}w^{6}+1148496z^{4}w^{8}+973824z^{2}w^{10}+320566w^{12}}{y^{12}-4y^{10}w^{2}-6y^{8}w^{4}+24y^{6}w^{6}+13y^{4}w^{8}-28y^{2}w^{10}-z^{12}-8z^{10}w^{2}-20z^{8}w^{4}-16z^{6}w^{6}-4z^{4}w^{8}+2w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.0.bs.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.24.0.m.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.24.1.h.2 | $16$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
16.96.3.a.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
16.96.3.bo.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
16.96.3.dn.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
16.96.3.dx.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
16.96.3.fj.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
16.96.3.fn.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
16.96.5.dp.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
16.96.5.dt.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.3.uj.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
48.96.3.un.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
48.96.3.vt.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
48.96.3.vx.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
48.96.3.yf.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
48.96.3.yj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
48.96.5.rj.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
48.96.5.rn.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
48.144.7.uc.1 | $48$ | $3$ | $3$ | $7$ | $4$ | $1^{2}\cdot2^{2}$ |
48.192.11.mk.1 | $48$ | $4$ | $4$ | $11$ | $4$ | $1^{8}\cdot2$ |
80.96.3.xt.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.xx.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.zd.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.zh.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.bcf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.3.bcj.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
80.96.5.sf.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.sj.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.240.17.kc.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
80.288.17.bem.1 | $80$ | $6$ | $6$ | $17$ | $?$ | not computed |
112.96.3.tj.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.tn.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.ut.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.ux.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.xf.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.xj.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.5.qr.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.qv.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.3.tj.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.3.tn.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.3.ut.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.3.ux.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.3.xf.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.3.xj.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
176.96.5.qr.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.qv.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.3.xt.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.3.xx.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.3.zd.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.3.zh.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.3.bcf.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.3.bcj.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
208.96.5.sf.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.sj.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.3.fjz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.fkd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.fmt.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.fmx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.frr.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.frv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.5.dot.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.dox.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.3.xt.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.xx.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.zd.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.zh.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.bcf.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.bcj.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.5.sf.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.sj.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.3.tj.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.3.tn.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.3.ut.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.3.ux.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.3.xf.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.3.xj.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
304.96.5.qr.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.qv.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |