Properties

Base field \(\Q(\zeta_{13})^+\)
Label 6.6.371293.1-53.2-a
Conductor 53.2
Rank \( 1 \)

Related objects

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Base field \(\Q(\zeta_{13})^+\)

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

Elliptic curves in class 53.2-a over \(\Q(\zeta_{13})^+\)

Isogeny class 53.2-a contains 2 curves linked by isogenies of degree 3.

Curve label Weierstrass Coefficients
53.2-a1 \( \bigl[a^{4} - 3 a^{2} + 1\) , \( a^{5} - 6 a^{3} + 8 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{4} - a^{3} + a^{2} + 2 a + 3\) , \( -a^{4} + 3 a^{2} - 1\bigr] \)
53.2-a2 \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 4 a^{2} + 3 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 3 a^{2} + 4 a + 1\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph