Properties

Base field 4.4.8069.1
Label 4.4.8069.1-19.1-c
Conductor 19.1
Rank bounds 0...1

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Base field 4.4.8069.1

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

Elliptic curves in class 19.1-c over 4.4.8069.1

Isogeny class 19.1-c contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
19.1-c1 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 85 a^{3} + 459 a^{2} - 366 a - 2135\) , \( -10862 a^{3} + 5071 a^{2} + 51661 a - 27097\bigr] \)
19.1-c2 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 40 a^{3} + 39 a^{2} - 181 a - 160\) , \( 261 a^{3} + 179 a^{2} - 1219 a - 762\bigr] \)
19.1-c3 \( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} + 11 a^{2} + 19 a - 45\) , \( -13 a^{3} + 30 a^{2} + 74 a - 109\bigr] \)
19.1-c4 \( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 4 a^{3} + 6 a^{2} - 11 a - 5\) , \( 7 a^{3} + 9 a^{2} - 18 a - 5\bigr] \)

Rank

Rank \(r\) satisfies \(0 \le r \le 1\)

Isogeny matrix

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

Isogeny graph