Properties

Base field \(\Q(\sqrt{30}) \)
Label 2.2.120.1-15.1-c
Number of curves 8
Graph
Conductor 15.1
Rank \( 1 \)

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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([0,0]),K([-106026,19358]),K([34806630,-6354792])]) 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-c have rank \( 1 \).

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-c 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-c contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
15.1-c1 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 19358 a - 106026\) , \( -6354792 a + 34806630\bigr] \)
15.1-c2 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -2 a + 14\) , \( 1448 a - 7930\bigr] \)
15.1-c3 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -6162 a + 33754\) , \( -356336 a + 1951734\bigr] \)
15.1-c4 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 1758 a - 9626\) , \( -35432 a + 194070\bigr] \)
15.1-c5 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 878 a - 4806\) , \( 37104 a - 203226\bigr] \)
15.1-c6 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 23758 a - 130126\) , \( -4553632 a + 24941270\bigr] \)
15.1-c7 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 14078 a - 77106\) , \( 2194824 a - 12021546\bigr] \)
15.1-c8 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 380158 a - 2082226\) , \( -296837272 a + 1625844710\bigr] \)