Properties

Base field 6.6.1279733.1
Label 6.6.1279733.1-41.1-b
Conductor 41.1
Rank \( 1 \)

Related objects

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Base field 6.6.1279733.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).

Elliptic curves in class 41.1-b over 6.6.1279733.1

Isogeny class 41.1-b contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
41.1-b1 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -10 a^{4} + 109 a^{3} - 59 a^{2} - 317 a - 71\) , \( -216 a^{4} + 1168 a^{3} - 88 a^{2} - 3288 a - 374\bigr] \)
41.1-b2 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 0\bigr] \)
41.1-b3 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a\) , \( 1\) , \( -6 a^{3} + 6 a^{2} + 18 a - 11\) , \( 4 a^{4} - 2 a^{3} - 18 a^{2} + 2 a + 16\bigr] \)
41.1-b4 \( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( -a^{4} + a^{3} + 4 a^{2} - 2 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 17 a^{4} - 4 a^{3} - 81 a^{2} - 5 a + 50\) , \( -25 a^{4} - 51 a^{3} + 176 a^{2} + 178 a - 273\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 10 & 5 & 2 \\ 10 & 1 & 2 & 5 \\ 5 & 2 & 1 & 10 \\ 2 & 5 & 10 & 1 \end{array}\right)\)

Isogeny graph