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Properties

Base field 3.3.148.1
Label 3.3.148.1-80.1-b
Number of curves 12
Graph
Conductor 80.1
Rank \( 0 \)

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Base field 3.3.148.1

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -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 \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([-1,0,1]),K([2,2,-1]),K([0,0,0]),K([2118,-5372,-4591]),K([-70406,178776,152788])]) 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 80.1-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 3 & 6 & 2 & 2 & 6 & 4 & 12 & 2 & 6 & 4 & 12 \\ 3 & 1 & 2 & 6 & 6 & 2 & 12 & 4 & 6 & 2 & 12 & 4 \\ 6 & 2 & 1 & 3 & 12 & 4 & 24 & 8 & 12 & 4 & 24 & 8 \\ 2 & 6 & 3 & 1 & 4 & 12 & 8 & 24 & 4 & 12 & 8 & 24 \\ 2 & 6 & 12 & 4 & 1 & 3 & 8 & 24 & 4 & 12 & 8 & 24 \\ 6 & 2 & 4 & 12 & 3 & 1 & 24 & 8 & 12 & 4 & 24 & 8 \\ 4 & 12 & 24 & 8 & 8 & 24 & 1 & 3 & 2 & 6 & 4 & 12 \\ 12 & 4 & 8 & 24 & 24 & 8 & 3 & 1 & 6 & 2 & 12 & 4 \\ 2 & 6 & 12 & 4 & 4 & 12 & 2 & 6 & 1 & 3 & 2 & 6 \\ 6 & 2 & 4 & 12 & 12 & 4 & 6 & 2 & 3 & 1 & 6 & 2 \\ 4 & 12 & 24 & 8 & 8 & 24 & 4 & 12 & 2 & 6 & 1 & 3 \\ 12 & 4 & 8 & 24 & 24 & 8 & 12 & 4 & 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 80.1-b over 3.3.148.1

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

Isogeny class 80.1-b contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
80.1-b1 \( \bigl[a^{2} - 1\) , \( -a^{2} + 2 a + 2\) , \( 0\) , \( -4591 a^{2} - 5372 a + 2118\) , \( 152788 a^{2} + 178776 a - 70406\bigr] \)
80.1-b2 \( \bigl[a^{2} - 1\) , \( -a^{2} + 2 a + 2\) , \( 0\) , \( -319191 a^{2} - 373482 a + 147088\) , \( 180297206 a^{2} + 210963324 a - 83082960\bigr] \)
80.1-b3 \( \bigl[a^{2} - 1\) , \( -a^{2} + 3\) , \( a^{2} - a - 2\) , \( -a^{2} + a - 7\) , \( 24 a^{2} - 17 a - 70\bigr] \)
80.1-b4 \( \bigl[a^{2} - 1\) , \( -a^{2} + 3\) , \( a^{2} - a - 2\) , \( -a^{2} + a + 3\) , \( -a^{2} + a + 3\bigr] \)
80.1-b5 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - a - 2\) , \( -746059 a^{2} - 872952 a + 343795\) , \( 645987240 a^{2} + 755860940 a - 297678109\bigr] \)
80.1-b6 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - a - 2\) , \( -60423904 a^{2} - 70701192 a + 27844010\) , \( 471004169433 a^{2} + 551115613798 a - 217043962228\bigr] \)
80.1-b7 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( 9483 a^{2} + 11097 a - 4368\) , \( -775196971 a^{2} - 907047500 a + 357219390\bigr] \)
80.1-b8 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( -85357 a^{2} - 99883 a + 39322\) , \( 20930333153 a^{2} + 24490299962 a - 9644930444\bigr] \)
80.1-b9 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( -414302 a^{2} - 484768 a + 190917\) , \( -262863548 a^{2} - 307573085 a + 121130448\bigr] \)
80.1-b10 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( -3792727 a^{2} - 4437818 a + 1747732\) , \( 7293172432 a^{2} + 8533642507 a - 3360775010\bigr] \)
80.1-b11 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( -6597627 a^{2} - 7719793 a + 3040262\) , \( -16993318815 a^{2} - 19883652708 a + 7830710398\bigr] \)
80.1-b12 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 1\) , \( -7759797 a^{2} - 9079633 a + 3575802\) , \( -10596995343 a^{2} - 12399401050 a + 4883213368\bigr] \)