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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-16250.5-a
Number of curves 8
Graph
Conductor 16250.5
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([1,0]),K([1,0]),K([1,1]),K([-5513,-4038]),K([124500,174937])]) 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 16250.5-a have rank \( 1 \).

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 16250.5-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 16250.5-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
16250.5-a1 \( \bigl[1\) , \( 1\) , \( i + 1\) , \( -4038 i - 5513\) , \( 174937 i + 124500\bigr] \)
16250.5-a2 \( \bigl[1\) , \( 1\) , \( i + 1\) , \( -38 i - 13\) , \( 437 i + 500\bigr] \)
16250.5-a3 \( \bigl[1\) , \( 1\) , \( i + 1\) , \( 11837 i + 3612\) , \( 1029062 i + 473125\bigr] \)
16250.5-a4 \( \bigl[i\) , \( -1\) , \( i + 1\) , \( 2712 i + 238\) , \( 21312 i + 34250\bigr] \)
16250.5-a5 \( \bigl[i\) , \( -1\) , \( i + 1\) , \( -4288 i - 5512\) , \( -158938 i - 129750\bigr] \)
16250.5-a6 \( \bigl[i\) , \( -1\) , \( i + 1\) , \( -24413 i - 14637\) , \( 1730187 i - 122375\bigr] \)
16250.5-a7 \( \bigl[i\) , \( -1\) , \( i + 1\) , \( 962 i - 12\) , \( -8438 i - 7500\bigr] \)
16250.5-a8 \( \bigl[1\) , \( 1\) , \( i + 1\) , \( 15212 i - 263\) , \( 530187 i + 497250\bigr] \)