↑

Properties

Base field \(\Q(\sqrt{14}) \)
Label 2.2.56.1-112.1-b
Number of curves 6
Graph
Conductor 112.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{14}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 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 \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-32996,-8820]),K([-3513640,-939060])]) 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 112.1-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 112.1-b over \(\Q(\sqrt{14}) \)

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

Isogeny class 112.1-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
112.1-b1 \( \bigl[a\) , \( 1\) , \( 0\) , \( -8820 a - 32996\) , \( -939060 a - 3513640\bigr] \)
112.1-b2 \( \bigl[a\) , \( 1\) , \( 0\) , \( -30 a + 119\) , \( 854 a - 3192\bigr] \)
112.1-b3 \( \bigl[a\) , \( 1\) , \( 0\) , \( 1770 a - 6616\) , \( -59376 a + 222168\bigr] \)
112.1-b4 \( \bigl[a\) , \( 1\) , \( 0\) , \( -570 a - 2126\) , \( -14340 a - 53652\bigr] \)
112.1-b5 \( \bigl[a\) , \( 1\) , \( 0\) , \( 8970 a - 33556\) , \( 907960 a - 3397272\bigr] \)
112.1-b6 \( \bigl[a\) , \( 1\) , \( 0\) , \( 8820 a - 32996\) , \( 939060 a - 3513640\bigr] \)