Properties

Base field 6.6.1241125.1
Label 6.6.1241125.1-45.1-a
Conductor 45.1
Rank \( 1 \)

Related objects

Learn more

Base field 6.6.1241125.1

Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).

Elliptic curves in class 45.1-a over 6.6.1241125.1

Isogeny class 45.1-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
45.1-a1 \( \bigl[-a^{5} + 7 a^{3} + 2 a^{2} - 10 a - 6\) , \( -3 a^{5} + a^{4} + 21 a^{3} - a^{2} - 34 a - 9\) , \( -3 a^{5} + 2 a^{4} + 20 a^{3} - 5 a^{2} - 30 a - 9\) , \( -54 a^{5} + 50 a^{4} + 320 a^{3} - 170 a^{2} - 377 a - 100\) , \( 194 a^{5} - 228 a^{4} - 1073 a^{3} + 851 a^{2} + 1049 a + 215\bigr] \)
45.1-a2 \( \bigl[-2 a^{5} + a^{4} + 14 a^{3} - a^{2} - 22 a - 10\) , \( a^{3} - 4 a - 2\) , \( a + 1\) , \( 11 a^{5} - a^{4} - 54 a^{3} - 21 a^{2} + 43 a + 20\) , \( -a^{5} + 31 a^{4} - 105 a^{2} - 82 a - 17\bigr] \)
45.1-a3 \( \bigl[-a^{5} + 7 a^{3} + 2 a^{2} - 10 a - 6\) , \( -3 a^{5} + a^{4} + 21 a^{3} - a^{2} - 34 a - 9\) , \( -3 a^{5} + 2 a^{4} + 20 a^{3} - 5 a^{2} - 30 a - 9\) , \( -224 a^{5} + 85 a^{4} + 1510 a^{3} - 80 a^{2} - 2302 a - 895\) , \( 1545 a^{5} - 573 a^{4} - 10410 a^{3} + 444 a^{2} + 15839 a + 6248\bigr] \)
45.1-a4 \( \bigl[-2 a^{5} + a^{4} + 14 a^{3} - a^{2} - 22 a - 10\) , \( a^{3} - 4 a - 2\) , \( a + 1\) , \( 6 a^{5} - a^{4} - 44 a^{3} - a^{2} + 73 a + 30\) , \( 5 a^{5} + 2 a^{4} - 31 a^{3} - 14 a^{2} + 35 a + 16\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph