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Properties

Base field \(\Q(\sqrt{93}) \)
Label 2.2.93.1-75.1-a
Number of curves 8
Graph
Conductor 75.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{93}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-23, -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 \( x^{2} - x - 23 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,0]),K([-41356,-9574]),K([9072429,2099208])]) 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 75.1-a 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 75.1-a over \(\Q(\sqrt{93}) \)

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

Isogeny class 75.1-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
75.1-a1 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -9574 a - 41356\) , \( 2099208 a + 9072429\bigr] \)
75.1-a2 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4 a + 4\) , \( -472 a - 2021\bigr] \)
75.1-a3 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 3041 a + 13164\) , \( 107608 a + 465082\bigr] \)
75.1-a4 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -874 a - 3756\) , \( 14008 a + 60559\bigr] \)
75.1-a5 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 439 a - 2316\) , \( 11311 a - 60156\bigr] \)
75.1-a6 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -11749 a - 50756\) , \( 1518008 a + 6560584\bigr] \)
75.1-a7 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -6964 a - 30076\) , \( -696072 a - 3008283\bigr] \)
75.1-a8 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -187924 a - 812156\) , \( 97324808 a + 420620839\bigr] \)