Properties

Base field 6.6.1312625.1
Label 6.6.1312625.1-4.1-a
Conductor 4.1
Rank \( 0 \)

Related objects

Learn more

Base field 6.6.1312625.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 7 x^{3} + 12 x^{2} - 12 x - 1 \); class number \(1\).

Elliptic curves in class 4.1-a over 6.6.1312625.1

Isogeny class 4.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
4.1-a1 \( \bigl[a^{5} - 6 a^{3} + 9 a\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a + 2\) , \( a^{5} - 6 a^{3} + a^{2} + 9 a - 3\) , \( 366 a^{5} + 192 a^{4} - 2302 a^{3} - 911 a^{2} + 3138 a + 245\) , \( 4708 a^{5} + 2285 a^{4} - 29436 a^{3} - 10909 a^{2} + 39749 a + 3127\bigr] \)
4.1-a2 \( \bigl[a + 1\) , \( -3 a^{5} - a^{4} + 20 a^{3} + 6 a^{2} - 29 a - 4\) , \( 2 a^{5} + a^{4} - 13 a^{3} - 4 a^{2} + 19 a + 1\) , \( -8 a^{5} - 5 a^{4} + 47 a^{3} + 21 a^{2} - 61 a - 8\) , \( -4 a^{5} - 2 a^{4} + 26 a^{3} + 10 a^{2} - 37 a - 4\bigr] \)
4.1-a3 \( \bigl[2 a^{5} + a^{4} - 13 a^{3} - 4 a^{2} + 20 a + 1\) , \( -2 a^{5} - a^{4} + 13 a^{3} + 5 a^{2} - 18 a - 4\) , \( a^{5} + a^{4} - 7 a^{3} - 5 a^{2} + 12 a + 4\) , \( 19 a^{5} + 5 a^{4} - 129 a^{3} - 25 a^{2} + 205 a + 19\) , \( -142 a^{5} - 30 a^{4} + 958 a^{3} + 168 a^{2} - 1501 a - 120\bigr] \)
4.1-a4 \( \bigl[-a^{5} + 7 a^{3} - 11 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 5 a^{2} + 9 a + 3\) , \( 2 a^{5} + a^{4} - 13 a^{3} - 4 a^{2} + 20 a\) , \( -a^{3} - 2 a^{2} + 2 a + 7\) , \( -2 a^{5} - a^{4} + 10 a^{3} - 13 a + 4\bigr] \)
4.1-a5 \( \bigl[a^{5} + a^{4} - 6 a^{3} - 4 a^{2} + 9 a + 1\) , \( a^{4} - 6 a^{2} - a + 6\) , \( 2 a^{5} + a^{4} - 13 a^{3} - 4 a^{2} + 20 a + 1\) , \( -3 a^{5} - 2 a^{4} + 23 a^{3} + 4 a^{2} - 39 a + 6\) , \( -a^{5} - 4 a^{4} + 14 a^{3} + 11 a^{2} - 31 a + 1\bigr] \)
4.1-a6 \( \bigl[2 a^{5} + a^{4} - 13 a^{3} - 4 a^{2} + 20 a\) , \( a^{4} - a^{3} - 6 a^{2} + 4 a + 5\) , \( a^{4} - 5 a^{2} + a + 5\) , \( 34 a^{5} - 31 a^{4} - 253 a^{3} + 247 a^{2} + 463 a - 494\) , \( -283 a^{5} + 332 a^{4} + 1924 a^{3} - 2255 a^{2} - 3101 a + 3671\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph