![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([1,1]),K([-1,-1]),K([1,1]),K([-140,-79]),K([-2881,-1664])])
E.isogeny_class()
![Copy content]()
sage:E.rank()
![Copy content]()
magma:Rank(E);
The elliptic curves in class 288.1-b have
rank \( 0 \).
![Copy content]()
sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 8 & 4 & 8 & 2 & 4 \\
8 & 1 & 2 & 4 & 4 & 8 \\
4 & 2 & 1 & 2 & 2 & 4 \\
8 & 4 & 2 & 1 & 4 & 8 \\
2 & 4 & 2 & 4 & 1 & 2 \\
4 & 8 & 4 & 8 & 2 & 1
\end{array}\right)\)
![Copy content]()
sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 288.1-b over \(\Q(\sqrt{3}) \)
![Copy content]()
sage:E.isogeny_class().curves
Isogeny class 288.1-b contains
6 curves linked by isogenies of
degrees dividing 8.
| Curve label |
Weierstrass Coefficients |
| 288.1-b1
| \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -79 a - 140\) , \( -1664 a - 2881\bigr] \)
|
| 288.1-b2
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 10822 a - 18744\) , \( 809620 a - 1402303\bigr] \)
|
| 288.1-b3
| \( \bigl[0\) , \( a\) , \( 0\) , \( 198 a - 342\) , \( 1746 a - 3024\bigr] \)
|
| 288.1-b4
| \( \bigl[0\) , \( a\) , \( 0\) , \( -668 a + 1158\) , \( 54226 a - 93922\bigr] \)
|
| 288.1-b5
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 11 a - 21\) , \( -20 a + 34\bigr] \)
|
| 288.1-b6
| \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -34 a + 24\) , \( -164 a + 214\bigr] \)
|
