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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-26000.3-b
Number of curves 8
Graph
Conductor 26000.3
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([1,0]),K([0,0]),K([-5,-665]),K([4725,-4950])]) 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 26000.3-b 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 26000.3-b over \(\Q(\sqrt{-1}) \)

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

Isogeny class 26000.3-b contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
26000.3-b1 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -665 i - 5\) , \( -4950 i + 4725\bigr] \)
26000.3-b2 \( \bigl[0\) , \( -i - 1\) , \( 0\) , \( 42 i + 56\) , \( -172 i + 179\bigr] \)
26000.3-b3 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 285 i + 145\) , \( -18130 i + 14965\bigr] \)
26000.3-b4 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -40 i - 5\) , \( -75 i + 100\bigr] \)
26000.3-b5 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 105 i + 135\) , \( -694 i + 567\bigr] \)
26000.3-b6 \( \bigl[0\) , \( -i - 1\) , \( 0\) , \( -158 i - 344\) , \( 308 i + 1039\bigr] \)
26000.3-b7 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 460 i + 745\) , \( -6375 i + 7500\bigr] \)
26000.3-b8 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 7355 i + 11885\) , \( -424194 i + 472567\bigr] \)