$\GL_2(\Z/16\Z)$-generators: |
$\begin{bmatrix}1&10\\0&1\end{bmatrix}$, $\begin{bmatrix}5&3\\8&9\end{bmatrix}$, $\begin{bmatrix}7&3\\0&1\end{bmatrix}$, $\begin{bmatrix}7&12\\0&7\end{bmatrix}$, $\begin{bmatrix}15&13\\8&15\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
16.96.1-16.v.2.1, 16.96.1-16.v.2.2, 16.96.1-16.v.2.3, 16.96.1-16.v.2.4, 16.96.1-16.v.2.5, 16.96.1-16.v.2.6, 16.96.1-16.v.2.7, 16.96.1-16.v.2.8, 16.96.1-16.v.2.9, 16.96.1-16.v.2.10, 16.96.1-16.v.2.11, 16.96.1-16.v.2.12, 32.96.1-16.v.2.1, 32.96.1-16.v.2.2, 32.96.1-16.v.2.3, 32.96.1-16.v.2.4, 32.96.1-16.v.2.5, 32.96.1-16.v.2.6, 32.96.1-16.v.2.7, 32.96.1-16.v.2.8, 48.96.1-16.v.2.1, 48.96.1-16.v.2.2, 48.96.1-16.v.2.3, 48.96.1-16.v.2.4, 48.96.1-16.v.2.5, 48.96.1-16.v.2.6, 48.96.1-16.v.2.7, 48.96.1-16.v.2.8, 48.96.1-16.v.2.9, 48.96.1-16.v.2.10, 48.96.1-16.v.2.11, 48.96.1-16.v.2.12, 80.96.1-16.v.2.1, 80.96.1-16.v.2.2, 80.96.1-16.v.2.3, 80.96.1-16.v.2.4, 80.96.1-16.v.2.5, 80.96.1-16.v.2.6, 80.96.1-16.v.2.7, 80.96.1-16.v.2.8, 80.96.1-16.v.2.9, 80.96.1-16.v.2.10, 80.96.1-16.v.2.11, 80.96.1-16.v.2.12, 96.96.1-16.v.2.1, 96.96.1-16.v.2.2, 96.96.1-16.v.2.3, 96.96.1-16.v.2.4, 96.96.1-16.v.2.5, 96.96.1-16.v.2.6, 96.96.1-16.v.2.7, 96.96.1-16.v.2.8, 112.96.1-16.v.2.1, 112.96.1-16.v.2.2, 112.96.1-16.v.2.3, 112.96.1-16.v.2.4, 112.96.1-16.v.2.5, 112.96.1-16.v.2.6, 112.96.1-16.v.2.7, 112.96.1-16.v.2.8, 112.96.1-16.v.2.9, 112.96.1-16.v.2.10, 112.96.1-16.v.2.11, 112.96.1-16.v.2.12, 160.96.1-16.v.2.1, 160.96.1-16.v.2.2, 160.96.1-16.v.2.3, 160.96.1-16.v.2.4, 160.96.1-16.v.2.5, 160.96.1-16.v.2.6, 160.96.1-16.v.2.7, 160.96.1-16.v.2.8, 176.96.1-16.v.2.1, 176.96.1-16.v.2.2, 176.96.1-16.v.2.3, 176.96.1-16.v.2.4, 176.96.1-16.v.2.5, 176.96.1-16.v.2.6, 176.96.1-16.v.2.7, 176.96.1-16.v.2.8, 176.96.1-16.v.2.9, 176.96.1-16.v.2.10, 176.96.1-16.v.2.11, 176.96.1-16.v.2.12, 208.96.1-16.v.2.1, 208.96.1-16.v.2.2, 208.96.1-16.v.2.3, 208.96.1-16.v.2.4, 208.96.1-16.v.2.5, 208.96.1-16.v.2.6, 208.96.1-16.v.2.7, 208.96.1-16.v.2.8, 208.96.1-16.v.2.9, 208.96.1-16.v.2.10, 208.96.1-16.v.2.11, 208.96.1-16.v.2.12, 224.96.1-16.v.2.1, 224.96.1-16.v.2.2, 224.96.1-16.v.2.3, 224.96.1-16.v.2.4, 224.96.1-16.v.2.5, 224.96.1-16.v.2.6, 224.96.1-16.v.2.7, 224.96.1-16.v.2.8, 240.96.1-16.v.2.1, 240.96.1-16.v.2.2, 240.96.1-16.v.2.3, 240.96.1-16.v.2.4, 240.96.1-16.v.2.5, 240.96.1-16.v.2.6, 240.96.1-16.v.2.7, 240.96.1-16.v.2.8, 240.96.1-16.v.2.9, 240.96.1-16.v.2.10, 240.96.1-16.v.2.11, 240.96.1-16.v.2.12, 272.96.1-16.v.2.1, 272.96.1-16.v.2.2, 272.96.1-16.v.2.3, 272.96.1-16.v.2.4, 272.96.1-16.v.2.5, 272.96.1-16.v.2.6, 272.96.1-16.v.2.7, 272.96.1-16.v.2.8, 272.96.1-16.v.2.9, 272.96.1-16.v.2.10, 272.96.1-16.v.2.11, 272.96.1-16.v.2.12, 304.96.1-16.v.2.1, 304.96.1-16.v.2.2, 304.96.1-16.v.2.3, 304.96.1-16.v.2.4, 304.96.1-16.v.2.5, 304.96.1-16.v.2.6, 304.96.1-16.v.2.7, 304.96.1-16.v.2.8, 304.96.1-16.v.2.9, 304.96.1-16.v.2.10, 304.96.1-16.v.2.11, 304.96.1-16.v.2.12 |
Cyclic 16-isogeny field degree: |
$2$ |
Cyclic 16-torsion field degree: |
$16$ |
Full 16-torsion field degree: |
$512$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 11x + 14 $ |
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.
Maps to other modular curves
$j$-invariant map
of degree 48 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{744x^{2}y^{14}-861743506x^{2}y^{12}z^{2}+3946501303224x^{2}y^{10}z^{4}-2404052725370889x^{2}y^{8}z^{6}+450301688638460016x^{2}y^{6}z^{8}-34266169928476438809x^{2}y^{4}z^{10}+1121366330130233819844x^{2}y^{2}z^{12}-13119880523481218023425x^{2}z^{14}-196876xy^{14}z+20032110312xy^{12}z^{3}-40645158196479xy^{10}z^{5}+16765017079652202xy^{8}z^{7}-2481637326273847576xy^{6}z^{9}+160992383698961342616xy^{4}z^{11}-4690996378190219311737xy^{2}z^{13}+50228506469524739981310xz^{15}-y^{16}+21481872y^{14}z^{2}-317842308708y^{12}z^{4}+319718459371944y^{10}z^{6}-82473813056611532y^{8}z^{8}+8224738401779468256y^{6}z^{10}-365602252934307716394y^{4}z^{12}+7147543060259635457184y^{2}z^{14}-47977490845124607868921z^{16}}{y^{2}(x^{2}y^{12}-4x^{2}y^{10}z^{2}-14x^{2}y^{8}z^{4}+16x^{2}y^{6}z^{6}+171x^{2}y^{4}z^{8}+28x^{2}y^{2}z^{10}+x^{2}z^{12}+14xy^{10}z^{3}-100xy^{8}z^{5}+106xy^{6}z^{7}+364xy^{4}z^{9}+57xy^{2}z^{11}+2xz^{13}-16y^{12}z^{2}+112y^{10}z^{4}-177y^{8}z^{6}+200y^{6}z^{8}-1108y^{4}z^{10}-192y^{2}z^{12}-7z^{14})}$ |
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.