Properties

Base field 4.4.13068.1
Label 4.4.13068.1-12.1-e
Conductor 12.1
Rank not recorded

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Base field 4.4.13068.1

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

Elliptic curves in class 12.1-e over 4.4.13068.1

Isogeny class 12.1-e contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
12.1-e1 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{3} - a^{2} - 6 a\) , \( -14 a^{3} - a^{2} + 33 a - 10\) , \( 72 a^{3} + 82 a^{2} - 86 a + 15\bigr] \)
12.1-e2 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( 2 a^{3} - 3 a^{2} - 11 a + 1\) , \( 0\) , \( 5 a^{3} - 19 a^{2} - 39 a + 10\) , \( 31 a^{3} + 32 a^{2} - 28 a + 3\bigr] \)
12.1-e3 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( a^{3} - 70 a^{2} + 223 a - 61\) , \( 858 a^{3} - 2138 a^{2} - 1689 a + 757\bigr] \)
12.1-e4 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( 61 a^{3} - 155 a^{2} - 112 a + 54\) , \( 1681 a^{3} - 3559 a^{2} - 5432 a + 2107\bigr] \)
12.1-e5 \( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( a^{2} - 3 a - 4\) , \( -a^{3} + 2 a^{2} + 5 a - 1\) , \( 98 a^{3} - 279 a^{2} - 84 a + 56\) , \( 1058 a^{3} - 2980 a^{2} - 895 a + 578\bigr] \)
12.1-e6 \( \bigl[-a^{3} + 2 a^{2} + 5 a - 1\) , \( -a^{2} + 3 a + 4\) , \( -a^{3} + 2 a^{2} + 5 a - 1\) , \( 18 a^{3} + 47 a^{2} - 253 a - 161\) , \( 197 a^{3} - 552 a^{2} - 175 a + 103\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph