Properties

Base field 5.5.65657.1
Label 5.5.65657.1-5.1-b
Conductor 5.1
Rank not recorded

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Base field 5.5.65657.1

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

Elliptic curves in class 5.1-b over 5.5.65657.1

Isogeny class 5.1-b contains 4 curves linked by isogenies of degrees dividing 15.

Curve label Weierstrass Coefficients
5.1-b1 \( \bigl[a^{2} - a - 1\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 10 a - 9\) , \( 0\) , \( 846 a^{4} - 1524 a^{3} - 3297 a^{2} + 4382 a + 1145\) , \( 20454 a^{4} - 36469 a^{3} - 77364 a^{2} + 102303 a + 27278\bigr] \)
5.1-b2 \( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 86 a^{4} + 73 a^{3} - 426 a^{2} - 515 a - 104\) , \( 1231 a^{4} + 500 a^{3} - 5653 a^{2} - 5184 a - 905\bigr] \)
5.1-b3 \( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 7\) , \( a^{4} - 5 a^{2} - a + 4\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( 5 a^{4} - 2 a^{3} - 25 a^{2} - 4 a + 29\) , \( a^{4} + 3 a^{3} - 10 a^{2} - 16 a + 22\bigr] \)
5.1-b4 \( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{4} - 2 a^{3} - 6 a^{2} + 5 a + 11\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 9\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)

Isogeny graph