Properties

Base field 3.3.148.1
Label 3.3.148.1-256.1-e
Number of curves 6
Graph
Conductor 256.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field 3.3.148.1

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -1, 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 \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([-1,0,1]),K([-2,-2,1]),K([0,0,0]),K([19941918,-50636292,-43275682]),K([71912117508,-182598448598,-156055514429])]) 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 256.1-e have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 256.1-e over 3.3.148.1

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

Isogeny class 256.1-e contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
256.1-e1 \( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 2\) , \( 0\) , \( -43275682 a^{2} - 50636292 a + 19941918\) , \( -156055514429 a^{2} - 182598448598 a + 71912117508\bigr] \)
256.1-e2 \( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 2\) , \( 0\) , \( -601076492 a^{2} - 703311482 a + 276982738\) , \( -14772849106763 a^{2} - 17285511108790 a + 6807493248682\bigr] \)
256.1-e3 \( \bigl[a^{2} - 1\) , \( a^{2} - a - 2\) , \( 0\) , \( 30 a^{2} - 10 a - 84\) , \( -110 a^{2} + 78 a + 348\bigr] \)
256.1-e4 \( \bigl[a^{2} - 1\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - 1\) , \( -631731729485826 a^{2} - 739180759844258 a + 291109010345492\) , \( 15618492416928874935446 a^{2} + 18274986918519657135933 a - 7197175095639575637344\bigr] \)
256.1-e5 \( \bigl[a^{2} - 1\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - 1\) , \( -10054032728703376 a^{2} - 11764087831951688 a + 4633010154510017\) , \( 1010923243394735375244094 a^{2} + 1182867626112340667548910 a - 465844679290399901815904\bigr] \)
256.1-e6 \( \bigl[0\) , \( a^{2} - a - 1\) , \( 0\) , \( 750561828568228721 a^{2} + 878222252985010170 a - 345867242247645683\) , \( 80171770176002958186701740874 a^{2} + 93807904891829134609275758426 a - 36944043783550265805016258622\bigr] \)