Properties

Base field \(\Q(\zeta_{21})^+\)
Label 6.6.453789.1-27.1-b
Conductor 27.1
Rank \( 1 \)

Related objects

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Base field \(\Q(\zeta_{21})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).

Elliptic curves in class 27.1-b over \(\Q(\zeta_{21})^+\)

Isogeny class 27.1-b contains 2 curves linked by isogenies of degree 13.

Curve label Weierstrass Coefficients
27.1-b1 \( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -391 a^{5} + 2607 a^{3} - 391 a^{2} - 3911 a + 522\) , \( -5979 a^{5} + 40423 a^{3} - 5979 a^{2} - 61479 a + 8968\bigr] \)
27.1-b2 \( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -a^{5} + 7 a^{3} - a^{2} - 11 a + 2\) , \( a^{5} - 7 a^{3} + a^{2} + 11 a - 2\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)

Isogeny graph