Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-b
Conductor 45.1
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 45.1-b over 5.5.65657.1

Isogeny class 45.1-b contains 4 curves linked by isogenies of degrees dividing 15.

Curve label Weierstrass Coefficients
45.1-b1 \( \bigl[-a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( -a^{4} + 5 a^{2} - 2\) , \( a^{3} - 4 a - 1\) , \( -527 a^{4} + 1330 a^{3} + 662 a^{2} - 1670 a - 937\) , \( 18278 a^{4} - 41440 a^{3} - 28524 a^{2} + 47105 a + 27660\bigr] \)
45.1-b2 \( \bigl[a\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\) , \( 330 a^{4} + 154 a^{3} - 1529 a^{2} - 1577 a - 314\) , \( 9329 a^{4} + 3425 a^{3} - 41530 a^{2} - 38029 a - 6558\bigr] \)
45.1-b3 \( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\) , \( 9 a^{3} - 3 a^{2} - 40 a - 8\) , \( 10 a^{4} + 2 a^{3} - 44 a^{2} - 35 a - 7\bigr] \)
45.1-b4 \( \bigl[a^{2} - a - 1\) , \( -a^{2} + a + 2\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 3\) , \( 4 a^{4} - 5 a^{3} - 19 a^{2} + 13 a + 17\) , \( 8 a^{4} - 10 a^{3} - 37 a^{2} + 25 a + 32\bigr] \)

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