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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-26000.3-f
Number of curves 8
Graph
Conductor 26000.3
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([-62,-5466]),K([107220,-110040])]) 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-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 26000.3-f over \(\Q(\sqrt{-1}) \)

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

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

Curve label Weierstrass Coefficients
26000.3-f1 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -5466 i - 62\) , \( -110040 i + 107220\bigr] \)
26000.3-f2 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( -26 i + 18\) , \( -408 i + 244\bigr] \)
26000.3-f3 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 7994 i - 5842\) , \( -464800 i + 663900\bigr] \)
26000.3-f4 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 1454 i - 1622\) , \( -26840 i + 10620\bigr] \)
26000.3-f5 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -5586 i + 98\) , \( 111688 i - 95284\bigr] \)
26000.3-f6 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( -21086 i + 8598\) , \( -135312 i + 1233716\bigr] \)
26000.3-f7 \( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 454 i - 622\) , \( 6360 i - 4980\bigr] \)
26000.3-f8 \( \bigl[i + 1\) , \( -i - 1\) , \( 0\) , \( 7134 i - 9862\) , \( -417928 i + 309604\bigr] \)