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Properties

Base field 6.6.1397493.1
Label 6.6.1397493.1-64.1-d
Number of curves 4
Graph
Conductor 64.1
Rank \( 1 \)

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Base field 6.6.1397493.1

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 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 \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0,0,0,0,0]),K([4,-6,-11,4,3,-1]),K([0,2,7,-2,-3,1]),K([-117,-2,-7,2,3,-1]),K([-672,236,431,-157,-117,39])]) 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 64.1-d have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 21 & 3 & 7 \\ 21 & 1 & 7 & 3 \\ 3 & 7 & 1 & 21 \\ 7 & 3 & 21 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 64.1-d over 6.6.1397493.1

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

Isogeny class 64.1-d contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
64.1-d1 \( \bigl[1\) , \( -a^{5} + 3 a^{4} + 4 a^{3} - 11 a^{2} - 6 a + 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} - 2 a - 117\) , \( 39 a^{5} - 117 a^{4} - 157 a^{3} + 431 a^{2} + 236 a - 672\bigr] \)
64.1-d2 \( \bigl[a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a - 1\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 11 a^{2} + 6 a - 6\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( -1\) , \( -4 a^{5} + 12 a^{4} + 14 a^{3} - 40 a^{2} - 20 a + 21\bigr] \)
64.1-d3 \( \bigl[a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a - 1\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 11 a^{2} + 6 a - 6\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( 40 a^{5} - 120 a^{4} - 130 a^{3} + 380 a^{2} + 180 a - 201\) , \( 346 a^{5} - 1038 a^{4} - 1162 a^{3} + 3362 a^{2} + 1632 a - 1777\bigr] \)
64.1-d4 \( \bigl[1\) , \( -a^{5} + 3 a^{4} + 4 a^{3} - 11 a^{2} - 6 a + 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} - 2 a + 3\) , \( -a^{5} + 3 a^{4} + 3 a^{3} - 9 a^{2} - 4 a + 2\bigr] \)