Properties

Base field 4.4.4205.1
Label 4.4.4205.1-49.3-d
Conductor 49.3
Rank \( 1 \)

Related objects

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Base field 4.4.4205.1

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

Elliptic curves in class 49.3-d over 4.4.4205.1

Isogeny class 49.3-d contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
49.3-d1 \( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( -3 a^{3} + 27 a^{2} - 18 a - 108\) , \( -238 a^{3} + 291 a^{2} + 1067 a + 79\bigr] \)
49.3-d2 \( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 8 a - 3\) , \( -6 a^{3} + 8 a^{2} + 25 a - 1\bigr] \)
49.3-d3 \( \bigl[-a^{3} + 2 a^{2} + 3 a - 1\) , \( a^{2} - 2 a - 2\) , \( 0\) , \( -17 a^{3} - 9 a^{2} + 19 a - 4\) , \( -79 a^{3} - 173 a^{2} - 46 a + 42\bigr] \)
49.3-d4 \( \bigl[-a^{3} + 2 a^{2} + 3 a\) , \( -a^{3} + 2 a^{2} + 2 a\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( -8 a^{3} + 12 a^{2} + 28 a + 6\) , \( -7 a^{3} + 8 a^{2} + 27 a + 19\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph