$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}3&9\\8&3\end{bmatrix}$, $\begin{bmatrix}3&14\\0&13\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$, $\begin{bmatrix}7&11\\0&3\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.96.1-16.s.2.1, 16.96.1-16.s.2.2, 16.96.1-16.s.2.3, 16.96.1-16.s.2.4, 16.96.1-16.s.2.5, 16.96.1-16.s.2.6, 16.96.1-16.s.2.7, 16.96.1-16.s.2.8, 48.96.1-16.s.2.1, 48.96.1-16.s.2.2, 48.96.1-16.s.2.3, 48.96.1-16.s.2.4, 48.96.1-16.s.2.5, 48.96.1-16.s.2.6, 48.96.1-16.s.2.7, 48.96.1-16.s.2.8, 80.96.1-16.s.2.1, 80.96.1-16.s.2.2, 80.96.1-16.s.2.3, 80.96.1-16.s.2.4, 80.96.1-16.s.2.5, 80.96.1-16.s.2.6, 80.96.1-16.s.2.7, 80.96.1-16.s.2.8, 112.96.1-16.s.2.1, 112.96.1-16.s.2.2, 112.96.1-16.s.2.3, 112.96.1-16.s.2.4, 112.96.1-16.s.2.5, 112.96.1-16.s.2.6, 112.96.1-16.s.2.7, 112.96.1-16.s.2.8, 176.96.1-16.s.2.1, 176.96.1-16.s.2.2, 176.96.1-16.s.2.3, 176.96.1-16.s.2.4, 176.96.1-16.s.2.5, 176.96.1-16.s.2.6, 176.96.1-16.s.2.7, 176.96.1-16.s.2.8, 208.96.1-16.s.2.1, 208.96.1-16.s.2.2, 208.96.1-16.s.2.3, 208.96.1-16.s.2.4, 208.96.1-16.s.2.5, 208.96.1-16.s.2.6, 208.96.1-16.s.2.7, 208.96.1-16.s.2.8, 240.96.1-16.s.2.1, 240.96.1-16.s.2.2, 240.96.1-16.s.2.3, 240.96.1-16.s.2.4, 240.96.1-16.s.2.5, 240.96.1-16.s.2.6, 240.96.1-16.s.2.7, 240.96.1-16.s.2.8, 272.96.1-16.s.2.1, 272.96.1-16.s.2.2, 272.96.1-16.s.2.3, 272.96.1-16.s.2.4, 272.96.1-16.s.2.5, 272.96.1-16.s.2.6, 272.96.1-16.s.2.7, 272.96.1-16.s.2.8, 304.96.1-16.s.2.1, 304.96.1-16.s.2.2, 304.96.1-16.s.2.3, 304.96.1-16.s.2.4, 304.96.1-16.s.2.5, 304.96.1-16.s.2.6, 304.96.1-16.s.2.7, 304.96.1-16.s.2.8 |
Cyclic 16-isogeny field degree: |
$2$ |
Cyclic 16-torsion field degree: |
$16$ |
Full 16-torsion field degree: |
$512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x z + y^{2} $ |
| $=$ | $35 x^{2} - 27 x z - 7 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 2 x^{2} y^{2} + 3 x^{2} z^{2} + z^{4} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{8}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^4\cdot7^2}\cdot\frac{907067769487360xz^{11}-2886681150947328xz^{9}w^{2}-931213649608704xz^{7}w^{4}+6319843755595776xz^{5}w^{6}+182112132867840xz^{3}w^{8}+64523764187568xzw^{10}-129574475923456z^{12}+298094008926208z^{10}w^{2}+461330347290624z^{8}w^{4}-1332552037533696z^{6}w^{6}+763060519376064z^{4}w^{8}+48251436231600z^{2}w^{10}-424023618123w^{12}}{w^{2}(1679616xz^{9}-1116857728xz^{7}w^{2}+1326273984xz^{5}w^{4}-280842912xz^{3}w^{6}+8297856xzw^{8}-559872z^{10}+157276624z^{8}w^{2}-56963872z^{6}w^{4}-51460584z^{4}w^{6}+8075592z^{2}w^{8}-64827w^{10})}$ |
Hi
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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.
This modular curve is minimally covered by the modular curves in the database listed below.