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Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-96.1-b
Number of curves 6
Graph
Conductor 96.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{3}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-168452,97256]),K([-37520076,21662226])]) 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 96.1-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 96.1-b over \(\Q(\sqrt{3}) \)

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

Isogeny class 96.1-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
96.1-b1 \( \bigl[0\) , \( 1\) , \( 0\) , \( 97256 a - 168452\) , \( 21662226 a - 37520076\bigr] \)
96.1-b2 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 9135 a - 15826\) , \( -619924 a + 1073738\bigr] \)
96.1-b3 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 570 a - 991\) , \( -9409 a + 16295\bigr] \)
96.1-b4 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 2720 a - 4711\) , \( -10125 a + 17537\bigr] \)
96.1-b5 \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -3 a - 12\) , \( -3 a + 16\bigr] \)
96.1-b6 \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -37569 a + 65072\) , \( -632444 a + 1095425\bigr] \)