Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-36288.7-c
Number of curves 8
Graph
Conductor 36288.7
Rank \( 0 \)

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Base field \(\Q(\sqrt{-7}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,1]),K([1,0]),K([1,1]),K([-2143,1530]),K([62306,-98073])]) 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 36288.7-c have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 36288.7-c over \(\Q(\sqrt{-7}) \)

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

Isogeny class 36288.7-c contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
36288.7-c1 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 1530 a - 2143\) , \( -98073 a + 62306\bigr] \)
36288.7-c2 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -45 a + 62\) , \( -45 a + 62\bigr] \)
36288.7-c3 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 180 a - 253\) , \( -279 a + 44\bigr] \)
36288.7-c4 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -111 a - 76\) , \( -833 a + 90\bigr] \)
36288.7-c5 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 1755 a - 2458\) , \( 43065 a - 29188\bigr] \)
36288.7-c6 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 51 a + 158\) , \( -621 a + 894\bigr] \)
36288.7-c7 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2205 a - 3088\) , \( -60219 a + 37304\bigr] \)
36288.7-c8 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 35280 a - 49393\) , \( -3873105 a + 2479562\bigr] \)