Properties

Base field 6.6.1397493.1
Label 6.6.1397493.1-19.2-a
Conductor 19.2
Rank \( 1 \)

Related objects

Learn more

Base field 6.6.1397493.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).

Elliptic curves in class 19.2-a over 6.6.1397493.1

Isogeny class 19.2-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
19.2-a1 \( \bigl[a^{2} - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{5} - 2 a^{4} - 6 a^{3} + 9 a^{2} + 9 a - 5\) , \( 61 a^{5} - 52 a^{4} - 294 a^{3} - 54 a^{2} + 119 a - 22\) , \( -385 a^{5} + 338 a^{4} + 1905 a^{3} + 207 a^{2} - 929 a + 163\bigr] \)
19.2-a2 \( \bigl[a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 10 a - 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 2 a - 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 3 a\) , \( -47 a^{5} + 97 a^{4} + 278 a^{3} - 375 a^{2} - 479 a + 89\) , \( 342 a^{5} - 795 a^{4} - 1606 a^{3} + 2521 a^{2} + 2619 a - 626\bigr] \)
19.2-a3 \( \bigl[a^{5} - 2 a^{4} - 5 a^{3} + 6 a^{2} + 8 a - 2\) , \( -a^{5} + 4 a^{4} - 11 a^{2} + 2 a + 3\) , \( a^{3} - a^{2} - 3 a\) , \( 5 a^{5} - 8 a^{4} - 20 a^{3} + 10 a^{2} + 12 a\) , \( 11 a^{5} - 10 a^{4} - 56 a^{3} - 2 a^{2} + 33 a - 6\bigr] \)
19.2-a4 \( \bigl[a^{2} - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{5} - 2 a^{4} - 6 a^{3} + 9 a^{2} + 9 a - 5\) , \( -14 a^{5} + 13 a^{4} + 71 a^{3} + a^{2} - 41 a + 8\) , \( -70 a^{5} + 64 a^{4} + 349 a^{3} + 20 a^{2} - 187 a + 31\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph