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Properties

Base field 3.3.148.1
Label 3.3.148.1-16.1-a
Number of curves 6
Graph
Conductor 16.1
Rank \( 0 \)

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Base field 3.3.148.1

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([-2,-1,1]),K([-3,-2,1]),K([-1,0,1]),K([-27,61,54]),K([-99,250,214])]) 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 16.1-a have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 16.1-a over 3.3.148.1

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

Isogeny class 16.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
16.1-a1 \( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 1\) , \( 54 a^{2} + 61 a - 27\) , \( 214 a^{2} + 250 a - 99\bigr] \)
16.1-a2 \( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 1\) , \( -16 a^{2} - 19 a + 8\) , \( 49 a^{2} + 57 a - 23\bigr] \)
16.1-a3 \( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 1\) , \( -226 a^{2} - 259 a + 103\) , \( 3566 a^{2} + 4170 a - 1643\bigr] \)
16.1-a4 \( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 3\) , \( a + 1\) , \( -119 a^{2} - 139 a + 58\) , \( -1195 a^{2} - 1399 a + 550\bigr] \)
16.1-a5 \( \bigl[a^{2} - 1\) , \( a^{2} - 1\) , \( 0\) , \( 2 a^{2} + 2 a - 1\) , \( 0\bigr] \)
16.1-a6 \( \bigl[a^{2} - 1\) , \( -a^{2} + a + 1\) , \( a^{2} - a - 2\) , \( 8858 a^{2} + 10364 a - 4082\) , \( -102778712 a^{2} - 120259982 a + 47361573\bigr] \)