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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-26000.4-j
Number of curves 8
Graph
Conductor 26000.4
Rank \( 1 \)

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Base field \(\Q(\sqrt{-1}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,0]),K([-637,-191]),K([6165,2970])]) 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 26000.4-j have rank \( 1 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 26000.4-j over \(\Q(\sqrt{-1}) \)

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

Isogeny class 26000.4-j contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
26000.4-j1 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -191 i - 637\) , \( 2970 i + 6165\bigr] \)
26000.4-j2 \( \bigl[0\) , \( i + 1\) , \( 0\) , \( 66 i + 24\) , \( -36 i - 235\bigr] \)
26000.4-j3 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 219 i + 233\) , \( 11702 i + 20389\bigr] \)
26000.4-j4 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -16 i - 37\) , \( 35 i + 120\bigr] \)
26000.4-j5 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 159 i + 63\) , \( 450 i + 775\bigr] \)
26000.4-j6 \( \bigl[0\) , \( i + 1\) , \( 0\) , \( -374 i - 56\) , \( 332 i - 759\bigr] \)
26000.4-j7 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 844 i + 233\) , \( 3327 i + 9264\bigr] \)
26000.4-j8 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 13469 i + 3733\) , \( 230702 i + 591639\bigr] \)