Properties

Base field 4.4.9248.1
Label 4.4.9248.1-16.4-a
Conductor 16.4
Rank \( 1 \)

Related objects

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Base field 4.4.9248.1

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

Elliptic curves in class 16.4-a over 4.4.9248.1

Isogeny class 16.4-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
16.4-a1 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 4\) , \( a\) , \( 3 a^{3} + 3 a^{2} - 5 a + 2\) , \( 2 a^{3} + 6 a^{2} + a - 4\bigr] \)
16.4-a2 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 4\) , \( a\) , \( 3 a^{3} - 7 a^{2} - 5 a + 7\) , \( -8 a^{3} + 16 a^{2} + 6 a - 9\bigr] \)
16.4-a3 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 4 a^{3} + 6 a^{2} - 9 a - 4\) , \( 5 a^{3} + 10 a^{2} - 4 a - 7\bigr] \)
16.4-a4 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( 424 a^{3} + 319 a^{2} - 1933 a - 1447\) , \( -7799 a^{3} - 5343 a^{2} + 35576 a + 24369\bigr] \)
16.4-a5 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -426 a^{3} + 319 a^{2} + 1947 a - 1447\) , \( 8435 a^{3} - 5713 a^{2} - 38472 a + 26069\bigr] \)
16.4-a6 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -a^{3} + 39 a^{2} + 7 a - 167\) , \( 38 a^{3} - 142 a^{2} - 168 a + 655\bigr] \)
16.4-a7 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -a^{3} + 4 a^{2} + 7 a - 7\) , \( 3 a^{3} + a^{2} - 8 a + 3\bigr] \)
16.4-a8 \( \bigl[a^{3} - 4 a\) , \( a^{2} - 2\) , \( 0\) , \( 5 a^{2} - 20\) , \( 7 a^{2} - 31\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph