Properties

Base field 4.4.17428.1
Label 4.4.17428.1-12.2-c
Conductor 12.2
Rank \( 0 \)

Related objects

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Base field 4.4.17428.1

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

Elliptic curves in class 12.2-c over 4.4.17428.1

Isogeny class 12.2-c contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
12.2-c1 \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -1\) , \( a^{3} - 3 a - 1\) , \( -a^{3} + 8 a^{2} - a - 16\) , \( 7 a^{3} - 11 a^{2} - 6 a + 9\bigr] \)
12.2-c2 \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( -36 a^{3} + 75 a^{2} + 172 a - 306\) , \( -358 a^{3} + 661 a^{2} + 1652 a - 2788\bigr] \)
12.2-c3 \( \bigl[a^{3} - 3 a - 1\) , \( a^{2} + a - 2\) , \( 0\) , \( 7 a^{3} - 11 a^{2} - 7 a + 6\) , \( -21 a^{3} + 77 a^{2} - 25 a - 78\bigr] \)
12.2-c4 \( \bigl[1\) , \( a^{2} - 2\) , \( a^{3} - 4 a\) , \( 8 a^{3} + 5 a^{2} - 42 a - 33\) , \( 26 a^{3} + 14 a^{2} - 136 a - 105\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph