Properties

Base field 6.6.1528713.1
Label 6.6.1528713.1-53.3-a
Conductor 53.3
Rank \( 0 \)

Related objects

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Base field 6.6.1528713.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 1 \); class number \(1\).

Elliptic curves in class 53.3-a over 6.6.1528713.1

Isogeny class 53.3-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
53.3-a1 \( \bigl[a^{5} - 3 a^{4} - 2 a^{3} + 4 a^{2} + 2 a + 2\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 2 a^{2} + 5 a + 2\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 3 a^{2} + 4 a + 1\) , \( 20 a^{5} - 70 a^{4} - 21 a^{3} + 133 a^{2} - 17 a - 18\) , \( -421 a^{5} + 1576 a^{4} + 104 a^{3} - 3058 a^{2} + 978 a + 593\bigr] \)
53.3-a2 \( \bigl[a^{5} - 2 a^{4} - 5 a^{3} + 2 a^{2} + 5 a + 2\) , \( a^{5} - 3 a^{4} - 3 a^{3} + 6 a^{2} + 4 a\) , \( 3 a^{5} - 10 a^{4} - 4 a^{3} + 18 a^{2} - a - 3\) , \( 9 a^{5} - 26 a^{4} - 28 a^{3} + 54 a^{2} + 26 a - 13\) , \( 9 a^{5} - 25 a^{4} - 31 a^{3} + 55 a^{2} + 28 a - 19\bigr] \)
53.3-a3 \( \bigl[2 a^{5} - 6 a^{4} - 5 a^{3} + 12 a^{2} + a - 2\) , \( -a^{3} + 3 a^{2} + 2 a - 4\) , \( 2 a^{5} - 7 a^{4} - 2 a^{3} + 14 a^{2} - 2 a - 4\) , \( 2 a^{5} - 5 a^{4} - 8 a^{3} + 3 a^{2} + 8 a - 3\) , \( 7 a^{5} - 16 a^{4} - 35 a^{3} + 22 a^{2} + 19 a - 8\bigr] \)
53.3-a4 \( \bigl[2 a^{5} - 5 a^{4} - 7 a^{3} + 7 a^{2} + 4 a + 2\) , \( a^{4} - 3 a^{3} - 3 a^{2} + 7 a + 2\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 2 a^{2} + 6 a + 2\) , \( 9 a^{5} + 7 a^{4} - 106 a^{3} - 59 a^{2} + 102 a + 14\) , \( 188 a^{5} - 397 a^{4} - 776 a^{3} + 308 a^{2} + 310 a + 11\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph