Properties

Base field 4.4.4752.1
Label 4.4.4752.1-33.1-f
Conductor 33.1
Rank \( 0 \)

Related objects

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Base field 4.4.4752.1

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

Elliptic curves in class 33.1-f over 4.4.4752.1

Isogeny class 33.1-f contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
33.1-f1 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( 335 a^{2} - 335 a - 1279\) , \( 4719 a^{2} - 4719 a - 17652\bigr] \)
33.1-f2 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( 40 a^{3} - 240 a^{2} + 370 a - 134\) , \( -1216 a^{3} + 4596 a^{2} - 3810 a + 141\bigr] \)
33.1-f3 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( 25 a^{2} - 25 a - 79\) , \( 89 a^{2} - 89 a - 316\bigr] \)
33.1-f4 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( 20 a^{2} - 20 a - 79\) , \( 84 a^{2} - 84 a - 306\bigr] \)
33.1-f5 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( -20 a^{2} + 20 a - 9\) , \( 100 a^{2} - 100 a - 16\bigr] \)
33.1-f6 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( -4\) , \( 4 a^{2} - 4 a - 7\bigr] \)
33.1-f7 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( 1\) , \( 0\bigr] \)
33.1-f8 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 1\) , \( 0\) , \( -40 a^{3} - 120 a^{2} - 10 a + 36\) , \( 1216 a^{3} + 948 a^{2} - 1734 a - 289\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 4 & 2 & 8 & 4 & 8 & 16 \\ 16 & 1 & 16 & 8 & 2 & 4 & 8 & 4 \\ 4 & 16 & 1 & 2 & 8 & 4 & 8 & 16 \\ 2 & 8 & 2 & 1 & 4 & 2 & 4 & 8 \\ 8 & 2 & 8 & 4 & 1 & 2 & 4 & 2 \\ 4 & 4 & 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 8 & 8 & 4 & 4 & 2 & 1 & 8 \\ 16 & 4 & 16 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph