Properties

Base field \(\Q(\sqrt{13}) \)
Label 2.2.13.1-1089.1-c
Number of curves 4
Graph
Conductor 1089.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{13}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -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 \( x^{2} - x - 3 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,1]),K([1,1]),K([5,-2]),K([-16,6])]) E.isogeny_class()
 

Rank

Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 

The elliptic curves in class 1089.1-c have rank \( 0 \).

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_class().graph().plot(edge_labels=True)
 

Elliptic curves in class 1089.1-c over \(\Q(\sqrt{13}) \)

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 

Isogeny class 1089.1-c contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
1089.1-c1 \( \bigl[1\) , \( a\) , \( a + 1\) , \( -2 a + 5\) , \( 6 a - 16\bigr] \)
1089.1-c2 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -97 a - 137\) , \( 141 a + 174\bigr] \)
1089.1-c3 \( \bigl[1\) , \( a\) , \( a + 1\) , \( 23 a - 55\) , \( 67 a - 157\bigr] \)
1089.1-c4 \( \bigl[1\) , \( a\) , \( a + 1\) , \( 78 a - 220\) , \( -549 a + 1328\bigr] \)