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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-26000.4-c
Number of curves 6
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([0,-1]),K([0,0]),K([13,23]),K([59,-2])]) 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-c have rank \( 1 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 26000.4-c 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-c contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
26000.4-c1 \( \bigl[i + 1\) , \( -i\) , \( 0\) , \( 23 i + 13\) , \( -2 i + 59\bigr] \)
26000.4-c2 \( \bigl[i + 1\) , \( -i\) , \( 0\) , \( -257 i - 197\) , \( 2392 i + 17\bigr] \)
26000.4-c3 \( \bigl[i + 1\) , \( -i\) , \( 0\) , \( -7 i + 53\) , \( 192 i + 117\bigr] \)
26000.4-c4 \( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 3 i - 2\) , \( 3 i - 2\bigr] \)
26000.4-c5 \( \bigl[i + 1\) , \( -i\) , \( 0\) , \( -237 i + 943\) , \( 10728 i + 4169\bigr] \)
26000.4-c6 \( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 373 i + 213\) , \( 12 i - 3489\bigr] \)