![Copy content]()
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
![Copy content]()
pari:K = nfinit(Polrev(%s));
![Copy content]()
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R!%s);
![Copy content]()
oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx(%s))
Generator \(a\), with minimal polynomial
\( x^{2} - 3 \); class number \(1\).
![Copy content]()
sage:E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-17512561,10110882]),K([39706205999,-22924388722])])
E.isogeny_class()
![Copy content]()
sage:E.rank()
![Copy content]()
magma:Rank(E);
The elliptic curves in class 3072.1-be have
rank \( 0 \).
![Copy content]()
sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 2 & 8 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
2 & 4 & 2 & 1 & 4 & 2 \\
8 & 4 & 2 & 4 & 1 & 8 \\
4 & 8 & 4 & 2 & 8 & 1
\end{array}\right)\)
![Copy content]()
sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 3072.1-be over \(\Q(\sqrt{3}) \)
![Copy content]()
sage:E.isogeny_class().curves
Isogeny class 3072.1-be contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 3072.1-be1
| \( \bigl[0\) , \( 1\) , \( 0\) , \( 10110882 a - 17512561\) , \( -22924388722 a + 39706205999\bigr] \)
|
| 3072.1-be2
| \( \bigl[0\) , \( 1\) , \( 0\) , \( 272778 a - 472465\) , \( 102151362 a - 176931349\bigr] \)
|
| 3072.1-be3
| \( \bigl[0\) , \( 1\) , \( 0\) , \( 17058 a - 29545\) , \( 1598538 a - 2768749\bigr] \)
|
| 3072.1-be4
| \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 282844 a - 489900\) , \( 13616268 a - 23584068\bigr] \)
|
| 3072.1-be5
| \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 48 a - 112\) , \( 1016 a - 1720\bigr] \)
|
| 3072.1-be6
| \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -1121676 a + 1942800\) , \( 105419284 a - 182591556\bigr] \)
|
