Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-45.1-a
Conductor 45.1
Rank not recorded

Related objects

Learn more about

Base field \(\Q(\zeta_{15})^+\)

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

Elliptic curves in class 45.1-a over \(\Q(\zeta_{15})^+\)

Isogeny class 45.1-a contains 10 curves linked by isogenies of degrees dividing 32.

Curve label Weierstrass Coefficients
45.1-a1 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -35 a^{3} + 174 a^{2} - 140 a - 32\) , \( -328 a^{3} + 1241 a^{2} - 884 a - 245\bigr] \)
45.1-a2 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - 5 a^{2} + 5 a\bigr] \)
45.1-a3 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 5 a^{3} - 26 a^{2} + 20 a + 8\) , \( 32 a^{3} - 129 a^{2} + 96 a + 25\bigr] \)
45.1-a4 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 10 a^{3} - 51 a^{2} + 40 a + 13\) , \( -49 a^{3} + 125 a^{2} - 65 a - 20\bigr] \)
45.1-a5 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 80 a^{3} - 401 a^{2} + 320 a + 83\) , \( 2177 a^{3} - 7659 a^{2} + 5241 a + 1450\bigr] \)
45.1-a6 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 110 a^{3} - 551 a^{2} + 440 a + 113\) , \( -6709 a^{3} + 22215 a^{2} - 14625 a - 4070\bigr] \)
45.1-a7 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 135 a^{3} - 676 a^{2} + 540 a + 138\) , \( -4874 a^{3} + 15925 a^{2} - 10390 a - 2895\bigr] \)
45.1-a8 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 2160 a^{3} - 10801 a^{2} + 8640 a + 2163\) , \( -309839 a^{3} + 1037335 a^{2} - 687955 a - 191220\bigr] \)
45.1-a9 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -17865 a^{3} - 26596 a^{2} + 58950 a + 11118\) , \( 1894864 a^{3} + 2845867 a^{2} - 6185020 a - 1357809\bigr] \)
45.1-a10 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 54585 a^{3} - 157006 a^{2} + 87930 a + 25608\) , \( -21999902 a^{3} + 64866343 a^{2} - 38895250 a - 11166531\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrr} 1 & 8 & 4 & 2 & 8 & 8 & 4 & 8 & 16 & 16 \\ 8 & 1 & 2 & 4 & 4 & 16 & 8 & 16 & 32 & 32 \\ 4 & 2 & 1 & 2 & 2 & 8 & 4 & 8 & 16 & 16 \\ 2 & 4 & 2 & 1 & 4 & 4 & 2 & 4 & 8 & 8 \\ 8 & 4 & 2 & 4 & 1 & 16 & 8 & 16 & 32 & 32 \\ 8 & 16 & 8 & 4 & 16 & 1 & 2 & 4 & 8 & 8 \\ 4 & 8 & 4 & 2 & 8 & 2 & 1 & 2 & 4 & 4 \\ 8 & 16 & 8 & 4 & 16 & 4 & 2 & 1 & 2 & 2 \\ 16 & 32 & 16 & 8 & 32 & 8 & 4 & 2 & 1 & 4 \\ 16 & 32 & 16 & 8 & 32 & 8 & 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph