Properties

Base field 6.6.1868969.1
Label 6.6.1868969.1-34.1-f
Conductor 34.1
Rank \( 1 \)

Related objects

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Base field 6.6.1868969.1

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

Elliptic curves in class 34.1-f over 6.6.1868969.1

Isogeny class 34.1-f contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
34.1-f1 \( \bigl[a^{5} - 5 a^{3} - a^{2} + 4 a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 5 a\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 3 a - 3\) , \( 1132 a^{5} - 847 a^{4} - 6478 a^{3} + 3274 a^{2} + 7402 a - 3761\) , \( -34273 a^{5} + 24088 a^{4} + 193967 a^{3} - 94014 a^{2} - 219967 a + 108548\bigr] \)
34.1-f2 \( \bigl[a^{5} + a^{4} - 5 a^{3} - 6 a^{2} + 3 a + 3\) , \( a^{4} - 4 a^{2} - a + 1\) , \( a^{5} - 5 a^{3} - a^{2} + 4 a\) , \( a^{5} - 5 a^{3} - a^{2} + 4 a\) , \( 0\bigr] \)
34.1-f3 \( \bigl[a^{4} - 5 a^{2} + 3\) , \( -a^{3} + 2 a + 2\) , \( a^{3} - 2 a - 1\) , \( -195 a^{5} + 259 a^{4} + 828 a^{3} - 908 a^{2} - 364 a + 299\) , \( -2784 a^{5} + 3671 a^{4} + 11862 a^{3} - 12858 a^{2} - 5314 a + 4222\bigr] \)
34.1-f4 \( \bigl[a^{5} - 4 a^{3} - a^{2} + a - 1\) , \( -a^{3} + 4 a\) , \( a^{3} - 3 a - 1\) , \( 2 a^{5} - 7 a^{4} - 25 a^{3} + 32 a^{2} + 60 a - 35\) , \( -75 a^{5} + 43 a^{4} + 366 a^{3} - 171 a^{2} - 320 a + 154\bigr] \)
34.1-f5 \( \bigl[a^{4} - 5 a^{2} + 3\) , \( -a^{3} + 2 a + 2\) , \( a^{3} - 2 a - 1\) , \( -3135 a^{5} + 4144 a^{4} + 13338 a^{3} - 14508 a^{2} - 5919 a + 4729\) , \( -167448 a^{5} + 220809 a^{4} + 713494 a^{3} - 773381 a^{2} - 319692 a + 253998\bigr] \)
34.1-f6 \( \bigl[a^{5} + a^{4} - 5 a^{3} - 5 a^{2} + 3 a + 1\) , \( 2 a^{4} - a^{3} - 8 a^{2} + 4\) , \( a^{5} + a^{4} - 5 a^{3} - 6 a^{2} + 4 a + 3\) , \( 119 a^{5} + 217 a^{4} - 572 a^{3} - 1103 a^{2} + 46 a + 296\) , \( -2014 a^{5} - 1950 a^{4} + 9286 a^{3} + 11188 a^{2} - 1230 a - 2838\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph