![Copy content]()
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, -1, 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
\( x^{2} - x + 4 \); class number \(2\).
![Copy content]()
sage:E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-162,69]),K([616,-573])])
E.isogeny_class()
![Copy content]()
sage:E.rank()
![Copy content]()
magma:Rank(E);
The elliptic curves in class 640.7-b have
rank \( 0 \).
![Copy content]()
sage:E.isogeny_class().matrix()
\(\left(\begin{array}{rrrrrrrr}
1 & 2 & 2 & 3 & 6 & 6 & 6 & 2 \\
2 & 1 & 4 & 6 & 12 & 12 & 3 & 4 \\
2 & 4 & 1 & 6 & 12 & 3 & 12 & 4 \\
3 & 6 & 6 & 1 & 2 & 2 & 2 & 6 \\
6 & 12 & 12 & 2 & 1 & 4 & 4 & 3 \\
6 & 12 & 3 & 2 & 4 & 1 & 4 & 12 \\
6 & 3 & 12 & 2 & 4 & 4 & 1 & 12 \\
2 & 4 & 4 & 6 & 3 & 12 & 12 & 1
\end{array}\right)\)
![Copy content]()
sage:E.isogeny_class().graph().plot(edge_labels=True)
![Copy content]()
sage:E.isogeny_class().curves
Isogeny class 640.7-b contains
8 curves linked by isogenies of
degrees dividing 12.
| Curve label |
Weierstrass Coefficients |
| 640.7-b1
| \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 69 a - 162\) , \( -573 a + 616\bigr] \)
|
| 640.7-b2
| \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 7 a + 2\) , \( 5 a - 49\bigr] \)
|
| 640.7-b3
| \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -4 a - 15\) , \( -a + 29\bigr] \)
|
| 640.7-b4
| \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -7 a + 16\) , \( 5 a - 13\bigr] \)
|
| 640.7-b5
| \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 23 a - 54\) , \( 13 a + 11\bigr] \)
|
| 640.7-b6
| \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -57 a + 166\) , \( 265 a + 507\bigr] \)
|
| 640.7-b7
| \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -72 a + 136\) , \( -60 a - 980\bigr] \)
|
| 640.7-b8
| \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 1149 a - 2682\) , \( -32325 a + 37840\bigr] \)
|
