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Properties

Base field \(\Q(\sqrt{6}) \)
Label 2.2.24.1-150.1-b
Number of curves 12
Graph
Conductor 150.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{6}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([-13776,-5606]),K([-969132,-395577])]) 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 150.1-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 24 & 8 & 3 & 6 & 8 & 4 & 12 & 12 & 2 & 24 & 4 \\ 24 & 1 & 3 & 8 & 4 & 12 & 6 & 8 & 2 & 12 & 4 & 24 \\ 8 & 3 & 1 & 24 & 12 & 4 & 2 & 24 & 6 & 4 & 12 & 8 \\ 3 & 8 & 24 & 1 & 2 & 24 & 12 & 4 & 4 & 6 & 8 & 12 \\ 6 & 4 & 12 & 2 & 1 & 12 & 6 & 2 & 2 & 3 & 4 & 6 \\ 8 & 12 & 4 & 24 & 12 & 1 & 2 & 24 & 6 & 4 & 3 & 8 \\ 4 & 6 & 2 & 12 & 6 & 2 & 1 & 12 & 3 & 2 & 6 & 4 \\ 12 & 8 & 24 & 4 & 2 & 24 & 12 & 1 & 4 & 6 & 8 & 3 \\ 12 & 2 & 6 & 4 & 2 & 6 & 3 & 4 & 1 & 6 & 2 & 12 \\ 2 & 12 & 4 & 6 & 3 & 4 & 2 & 6 & 6 & 1 & 12 & 2 \\ 24 & 4 & 12 & 8 & 4 & 3 & 6 & 8 & 2 & 12 & 1 & 24 \\ 4 & 24 & 8 & 12 & 6 & 8 & 4 & 3 & 12 & 2 & 24 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 150.1-b over \(\Q(\sqrt{6}) \)

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

Isogeny class 150.1-b contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
150.1-b1 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -5606 a - 13776\) , \( -395577 a - 969132\bigr] \)
150.1-b2 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 271 a - 661\) , \( -12352 a + 30257\bigr] \)
150.1-b3 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -29 a + 74\) , \( 416 a - 1018\bigr] \)
150.1-b4 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 35919 a + 87939\) , \( 803878 a + 1968921\bigr] \)
150.1-b5 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -9071 a - 22221\) , \( 98592 a + 241497\bigr] \)
150.1-b6 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 1371 a - 3356\) , \( -36992 a + 90612\bigr] \)
150.1-b7 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -371 a - 906\) , \( -5568 a - 13638\bigr] \)
150.1-b8 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -35919 a + 87939\) , \( -803878 a + 1968921\bigr] \)
150.1-b9 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -6671 a - 16341\) , \( 462144 a + 1132017\bigr] \)
150.1-b10 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -5771 a - 14136\) , \( -374496 a - 917328\bigr] \)
150.1-b11 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -106671 a - 261341\) , \( 29666144 a + 72667017\bigr] \)
150.1-b12 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 5606 a - 13776\) , \( 395577 a - 969132\bigr] \)