Properties

Base field \(\Q(\sqrt{30}) \)
Label 2.2.120.1-15.1-b
Number of curves 8
Graph
Conductor 15.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{30}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-30, 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 \( x^{2} - 30 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-51120511,9333283]),K([373348109476,-68163727122])]) 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 15.1-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 15.1-b over \(\Q(\sqrt{30}) \)

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

Isogeny class 15.1-b contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
15.1-b1 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 9333283 a - 51120511\) , \( -68163727122 a + 373348109476\bigr] \)
15.1-b2 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 1763 a - 9671\) , \( 15245718 a - 83504244\bigr] \)
15.1-b3 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( -2967357 a + 16252869\) , \( -3478975110 a + 19055131440\bigr] \)
15.1-b4 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 850083 a - 4656111\) , \( -458404002 a + 2510782116\bigr] \)
15.1-b5 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 425923 a - 2332891\) , \( 350864730 a - 1921765280\bigr] \)
15.1-b6 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 11454083 a - 62736611\) , \( -49312315902 a + 270094677816\bigr] \)
15.1-b7 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 6788323 a - 37181191\) , \( 22558466070 a - 123557807300\bigr] \)
15.1-b8 \( \bigl[a\) , \( -a - 1\) , \( a\) , \( 183238883 a - 1003640711\) , \( -3159143370282 a + 17303340862956\bigr] \)