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Properties

Base field 6.6.1397493.1
Label 6.6.1397493.1-64.1-a
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([-6,9,9,-6,-2,1]),K([0,-4,-1,1,0,0]),K([0,1,0,0,0,0]),K([4,-2,-126,-237,-117,-1]),K([-313,1885,-1226,-2271,3324,2525])]) 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-a 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-a 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-a contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
64.1-a1 \( \bigl[a^{5} - 2 a^{4} - 6 a^{3} + 9 a^{2} + 9 a - 6\) , \( a^{3} - a^{2} - 4 a\) , \( a\) , \( -a^{5} - 117 a^{4} - 237 a^{3} - 126 a^{2} - 2 a + 4\) , \( 2525 a^{5} + 3324 a^{4} - 2271 a^{3} - 1226 a^{2} + 1885 a - 313\bigr] \)
64.1-a2 \( \bigl[a^{5} - 3 a^{4} - 3 a^{3} + 9 a^{2} + 5 a - 3\) , \( -a^{4} + 4 a^{3} - 8 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{5} + 2 a^{4} + 10 a^{3} - 12 a^{2} - 24 a + 3\) , \( a^{5} + a^{4} - 5 a^{3} - 10 a^{2} - 7 a - 4\bigr] \)
64.1-a3 \( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 11 a^{2} + 6 a - 5\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + a - 2\) , \( -85 a^{5} + 97 a^{4} + 338 a^{3} - 237 a^{2} - 347 a + 77\) , \( 2530 a^{5} - 1383 a^{4} - 9879 a^{3} + 1347 a^{2} + 6782 a - 1319\bigr] \)
64.1-a4 \( \bigl[a^{5} - 3 a^{4} - 3 a^{3} + 10 a^{2} + 4 a - 4\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 5 a^{2} - 10 a - 1\) , \( a^{2} - a - 2\) , \( 12 a^{5} - 30 a^{4} - 52 a^{3} + 94 a^{2} + 87 a - 23\) , \( -23 a^{5} + 55 a^{4} + 102 a^{3} - 168 a^{2} - 169 a + 38\bigr] \)