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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-26000.4-f
Number of curves 8
Graph
Conductor 26000.4
Rank \( 0 \)

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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([-5230,-1590]),K([-139092,-65256])]) 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-f have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 26000.4-f 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-f contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
26000.4-f1 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -1590 i - 5230\) , \( -65256 i - 139092\bigr] \)
26000.4-f2 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 10 i - 30\) , \( -296 i - 372\bigr] \)
26000.4-f3 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -3370 i + 9310\) , \( -201360 i - 785020\bigr] \)
26000.4-f4 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -1150 i + 1850\) , \( -21384 i - 19388\bigr] \)
26000.4-f5 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -1470 i - 5390\) , \( 71000 i + 128500\bigr] \)
26000.4-f6 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 2350 i - 22650\) , \( 307616 i - 1202388\bigr] \)
26000.4-f7 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -470 i + 610\) , \( 4200 i + 6900\bigr] \)
26000.4-f8 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -7470 i + 9610\) , \( -282200 i - 436900\bigr] \)