Properties

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

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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([-3,0,1]),K([-1,0,1]),K([39659693498548,-100703452430598,-86064964924926]),K([361911795882278631324,-918962405043287659087,-785379897594692293591])]) 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-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 256.1-b 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-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
256.1-b1 \( \bigl[a^{2} - 1\) , \( a^{2} - 3\) , \( a^{2} - 1\) , \( -86064964924926 a^{2} - 100703452430598 a + 39659693498548\) , \( -785379897594692293591 a^{2} - 918962405043287659087 a + 361911795882278631324\bigr] \)
256.1-b2 \( \bigl[a^{2} - 1\) , \( a^{2} - 3\) , \( a^{2} - 1\) , \( -1369726948580196 a^{2} - 1602698992901198 a + 631185419116543\) , \( -50834524782266614880552 a^{2} - 59480790501786385311994 a + 23425114665039070310143\bigr] \)
256.1-b3 \( \bigl[a^{2} - 1\) , \( a^{2} - 2\) , \( 0\) , \( -10 a^{2} + 46 a - 36\) , \( 107 a^{2} - 286 a + 136\bigr] \)
256.1-b4 \( \bigl[a^{2} - 1\) , \( 0\) , \( 0\) , \( -5895730 a^{2} - 6898514 a + 2716818\) , \( 7839217703 a^{2} + 9172562700 a - 3612398746\bigr] \)
256.1-b5 \( \bigl[a^{2} - 1\) , \( 0\) , \( 0\) , \( -81888600 a^{2} - 95816744 a + 37735178\) , \( 742744202441 a^{2} + 869074954296 a - 342264793140\bigr] \)
256.1-b6 \( \bigl[0\) , \( a^{2} - a - 2\) , \( 0\) , \( 102253970213414727 a^{2} + 119645988750577542 a - 47119767273605408\) , \( -4031456994839579610987491418 a^{2} - 4717153351075791871135103011 a + 1857740242006452258727417755\bigr] \)