![Copy content]()
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 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
\( x^{2} - 19 \); class number \(1\).
![Copy content]()
sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-169,0]),K([535,0])])
E.isogeny_class()
![Copy content]()
sage:E.rank()
![Copy content]()
magma:Rank(E);
The elliptic curves in class 98.1-b have
rank \( 0 \).
![Copy content]()
sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrr}
1 & 9 & 3 & 6 & 18 & 2 \\
9 & 1 & 3 & 6 & 2 & 18 \\
3 & 3 & 1 & 2 & 6 & 6 \\
6 & 6 & 2 & 1 & 3 & 3 \\
18 & 2 & 6 & 3 & 1 & 9 \\
2 & 18 & 6 & 3 & 9 & 1
\end{array}\right)\)
![Copy content]()
sage:E.isogeny_class().graph().plot(edge_labels=True)
Elliptic curves in class 98.1-b over \(\Q(\sqrt{19}) \)
![Copy content]()
sage:E.isogeny_class().curves
Isogeny class 98.1-b contains
6 curves linked by isogenies of
degrees dividing 18.
| Curve label |
Weierstrass Coefficients |
| 98.1-b1
| \( \bigl[a\) , \( 1\) , \( a\) , \( -169\) , \( 535\bigr] \)
|
| 98.1-b2
| \( \bigl[a\) , \( 1\) , \( a\) , \( 1\) , \( 1\bigr] \)
|
| 98.1-b3
| \( \bigl[a\) , \( 1\) , \( a\) , \( 6\) , \( 17\bigr] \)
|
| 98.1-b4
| \( \bigl[a\) , \( 1\) , \( a\) , \( -34\) , \( 1\bigr] \)
|
| 98.1-b5
| \( \bigl[a\) , \( 1\) , \( a\) , \( -9\) , \( -31\bigr] \)
|
| 98.1-b6
| \( \bigl[a\) , \( 1\) , \( a\) , \( -2729\) , \( 49687\bigr] \)
|
