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Properties

Base field \(\Q(\zeta_{13})^+\)
Label 6.6.371293.1-79.4-c
Number of curves 4
Graph
Conductor 79.4
Rank \( 0 \)

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Base field \(\Q(\zeta_{13})^+\)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([3,2,-4,-4,1,1]),K([2,-2,-4,1,1,0]),K([-2,1,1,0,0,0]),K([11,9,-44,16,13,-6]),K([-14,-30,107,-38,-33,15])]) 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 79.4-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 10 & 5 \\ 2 & 1 & 5 & 10 \\ 10 & 5 & 1 & 2 \\ 5 & 10 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 79.4-c over \(\Q(\zeta_{13})^+\)

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

Isogeny class 79.4-c contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
79.4-c1 \( \bigl[a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( a^{2} + a - 2\) , \( -6 a^{5} + 13 a^{4} + 16 a^{3} - 44 a^{2} + 9 a + 11\) , \( 15 a^{5} - 33 a^{4} - 38 a^{3} + 107 a^{2} - 30 a - 14\bigr] \)
79.4-c2 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{4} + a^{3} - 3 a^{2} - 4 a\) , \( -2 a^{5} + a^{4} + 12 a^{3} - a^{2} - 16 a - 5\bigr] \)
79.4-c3 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 4 a^{2} - 9 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( 8 a^{5} + 87 a^{4} - 136 a^{3} - 266 a^{2} + 327 a - 71\) , \( -465 a^{5} + 1504 a^{4} + 575 a^{3} - 4782 a^{2} + 2674 a + 6\bigr] \)
79.4-c4 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 4 a^{2} - 9 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( -947 a^{5} + 2047 a^{4} + 2349 a^{3} - 6681 a^{2} + 2027 a + 739\) , \( -37043 a^{5} + 78539 a^{4} + 95061 a^{3} - 254944 a^{2} + 69625 a + 31719\bigr] \)