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Properties

Base field \(\Q(\zeta_{24})^+\)
Label 4.4.2304.1-64.1-b
Number of curves 8
Graph
Conductor 64.1
Rank \( 1 \)

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Base field \(\Q(\zeta_{24})^+\)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 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 \( x^{4} - 4 x^{2} + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([-1,-3,1,1]),K([-2,-1,1,0]),K([0,0,0,0]),K([10,19,-1,-5]),K([-4,-7,2,2])]) 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-b have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 2 & 8 & 2 \\ 4 & 2 & 4 & 1 & 2 & 8 & 2 & 8 \\ 8 & 4 & 8 & 2 & 1 & 16 & 4 & 16 \\ 8 & 4 & 2 & 8 & 16 & 1 & 16 & 4 \\ 8 & 4 & 8 & 2 & 4 & 16 & 1 & 16 \\ 8 & 4 & 2 & 8 & 16 & 4 & 16 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 64.1-b over \(\Q(\zeta_{24})^+\)

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

Isogeny class 64.1-b contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
64.1-b1 \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} - a - 2\) , \( 0\) , \( -5 a^{3} - a^{2} + 19 a + 10\) , \( 2 a^{3} + 2 a^{2} - 7 a - 4\bigr] \)
64.1-b2 \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} - a - 2\) , \( 0\) , \( -a^{3} - a^{2} - a\) , \( -2 a^{3} - 3 a^{2} + a + 1\bigr] \)
64.1-b3 \( \bigl[a^{3} - 3 a\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \)
64.1-b4 \( \bigl[a^{3} - 3 a\) , \( 1\) , \( a^{3} - 3 a\) , \( -3\) , \( 0\bigr] \)
64.1-b5 \( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 1\) , \( -16 a^{3} + 3 a^{2} + 63 a - 28\) , \( 19 a^{3} + 21 a^{2} - 146 a + 68\bigr] \)
64.1-b6 \( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 5 a - 1\) , \( a^{3} - 4 a + 1\) , \( -15 a^{3} + 5 a^{2} + 58 a - 31\) , \( -35 a^{3} - 17 a^{2} + 209 a - 99\bigr] \)
64.1-b7 \( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a + 1\) , \( 14 a^{3} + 3 a^{2} - 59 a - 29\) , \( -16 a^{3} + 20 a^{2} + 117 a + 53\bigr] \)
64.1-b8 \( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 5 a - 1\) , \( 0\) , \( 14 a^{3} + 5 a^{2} - 61 a - 30\) , \( 30 a^{3} - 16 a^{2} - 177 a - 83\bigr] \)