Properties

Base field 6.6.1387029.1
Label 6.6.1387029.1-25.1-b
Conductor 25.1
Rank \( 1 \)

Related objects

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Base field 6.6.1387029.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).

Elliptic curves in class 25.1-b over 6.6.1387029.1

Isogeny class 25.1-b contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
25.1-b1 \( \bigl[a^{4} - a^{3} - 3 a^{2} + a + 1\) , \( -a^{5} + 3 a^{4} + 3 a^{3} - 10 a^{2} - 3 a + 4\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} - a + 1\) , \( -403 a^{5} + 929 a^{4} + 1453 a^{3} - 2587 a^{2} - 1411 a + 546\) , \( 2770 a^{5} - 6296 a^{4} - 10066 a^{3} + 17554 a^{2} + 9875 a - 3777\bigr] \)
25.1-b2 \( \bigl[a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2} - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 6 a + 2\) , \( a\) , \( -2 a^{5} + a^{4} + 13 a^{3} + 6 a^{2} - 10 a\) , \( -6 a^{5} + 7 a^{4} + 26 a^{3} - 11 a^{2} - 14 a + 5\bigr] \)
25.1-b3 \( \bigl[-2 a^{5} + 5 a^{4} + 7 a^{3} - 14 a^{2} - 6 a + 5\) , \( a^{3} - 2 a^{2} - a + 1\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2}\) , \( 202 a^{5} - 536 a^{4} - 559 a^{3} + 1623 a^{2} + 258 a - 719\) , \( 1894 a^{5} - 5108 a^{4} - 5209 a^{3} + 15443 a^{2} + 2383 a - 6846\bigr] \)
25.1-b4 \( \bigl[a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2} - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 6 a + 2\) , \( a\) , \( 3 a^{5} - 29 a^{4} + 3 a^{3} + 96 a^{2} - 5 a - 40\) , \( -73 a^{5} + 95 a^{4} + 223 a^{3} - 264 a^{2} - 102 a + 116\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 6 & 2 & 3 \\ 6 & 1 & 3 & 2 \\ 2 & 3 & 1 & 6 \\ 3 & 2 & 6 & 1 \end{array}\right)\)

Isogeny graph