Properties

Base field 6.6.300125.1
Label 6.6.300125.1-71.5-c
Conductor 71.5
Rank \( 0 \)

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Base field 6.6.300125.1

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

Elliptic curves in class 71.5-c over 6.6.300125.1

Isogeny class 71.5-c contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
71.5-c1 \( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 39 a - 12\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 21 a - 6\) , \( 1\) , \( -944 a^{5} + 1869 a^{4} + 4540 a^{3} - 6026 a^{2} + 652 a + 593\) , \( -34009 a^{5} + 72266 a^{4} + 157382 a^{3} - 246003 a^{2} + 35333 a + 30593\bigr] \)
71.5-c2 \( \bigl[-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a - 1\) , \( -7 a^{5} + a^{4} + 51 a^{3} + 28 a^{2} - 32 a - 13\) , \( -a^{5} + 8 a^{3} + 4 a^{2} - 6 a - 1\) , \( 5 a^{5} - 38 a^{3} - 22 a^{2} + 26 a + 9\) , \( -3 a^{5} + a^{4} + 21 a^{3} + 8 a^{2} - 12 a - 4\bigr] \)
71.5-c3 \( \bigl[a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a\) , \( -4 a^{5} + 30 a^{3} + 19 a^{2} - 18 a - 9\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 6\) , \( -36 a^{5} - 3 a^{4} + 278 a^{3} + 176 a^{2} - 156 a - 75\) , \( 1119 a^{5} - 361 a^{4} - 7802 a^{3} - 3654 a^{2} + 4623 a + 1358\bigr] \)
71.5-c4 \( \bigl[-2 a^{5} + 15 a^{3} + 10 a^{2} - 9 a - 5\) , \( a^{5} - 7 a^{3} - 5 a^{2} + a + 3\) , \( a + 1\) , \( 1534 a^{5} - 351 a^{4} - 11009 a^{3} - 5421 a^{2} + 6557 a + 1992\) , \( -18999 a^{5} + 4356 a^{4} + 136350 a^{3} + 67090 a^{2} - 81287 a - 24651\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph