Properties

Base field 3.3.148.1
Label 3.3.148.1-8.1-a
Number of curves 6
Graph
Conductor 8.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

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([-1,0,0]),K([-2,-1,1]),K([-795941171665179475905777,2021044966504285468600887,1727261180779898112178031]),K([-128973872132688546804017272390232511682,327489021002657174715208924802606509852,279884457042981407569057081184688558284])]) 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 8.1-a have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 8.1-a over 3.3.148.1

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

Isogeny class 8.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
8.1-a1 \( \bigl[a^{2} - 1\) , \( -1\) , \( a^{2} - a - 2\) , \( 1727261180779898112178031 a^{2} + 2021044966504285468600887 a - 795941171665179475905777\) , \( 279884457042981407569057081184688558284 a^{2} + 327489021002657174715208924802606509852 a - 128973872132688546804017272390232511682\bigr] \)
8.1-a2 \( \bigl[a^{2} - 1\) , \( -a^{2} + a + 1\) , \( 0\) , \( -122876240093597390564 a^{2} - 143775828060934351842 a + 56622738702290433353\) , \( 1339806177584198660985008156380 a^{2} + 1567689103089350829411708766103 a - 617397594907508652287429548778\bigr] \)
8.1-a3 \( \bigl[a + 1\) , \( a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( 4 a^{2} + 15 a - 2\) , \( -14 a^{2} + 6 a - 2\bigr] \)
8.1-a4 \( \bigl[a + 1\) , \( a^{2} - 2 a - 1\) , \( a^{2} - a - 2\) , \( -83347907948 a^{2} - 97524260781 a + 38407643413\) , \( -13188165606368355 a^{2} - 15431294359397846 a + 6077253458627265\bigr] \)
8.1-a5 \( \bigl[a + 1\) , \( a^{2} - 2 a - 1\) , \( a^{2} - a - 2\) , \( -1157658655878 a^{2} - 1354560749371 a + 533461990118\) , \( -1248619261868812725 a^{2} - 1460992525253670641 a + 575377649491803209\bigr] \)
8.1-a6 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 1\) , \( -23137255028952366700772 a^{2} - 27072589446997404913200 a + 10661904569953464468393\) , \( 3529193720251159834920437017851122 a^{2} + 4129461880751340931496180433614334 a - 1626291736297668628974848142500566\bigr] \)