Properties

Base field 4.4.13068.1
Label 4.4.13068.1-12.1-d
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-d over 4.4.13068.1

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

Curve label Weierstrass Coefficients
12.1-d1 \( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - a - 2\) , \( 84 a^{3} - 148 a^{2} - 459 a + 156\) , \( -490 a^{3} + 763 a^{2} + 2533 a - 894\bigr] \)
12.1-d2 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{2} - 3 a - 3\) , \( 0\) , \( 23 a^{3} - 9 a^{2} - 157 a - 92\) , \( -221 a^{3} + 158 a^{2} + 1358 a + 633\bigr] \)
12.1-d3 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( 2 a^{3} - 3 a^{2} - 10 a\) , \( a + 1\) , \( 67 a^{3} - 53 a^{2} - 401 a - 178\) , \( -500 a^{3} + 346 a^{2} + 3081 a + 1511\bigr] \)
12.1-d4 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( -159 a^{3} + 227 a^{2} + 886 a - 312\) , \( -2178 a^{3} + 3457 a^{2} + 10932 a - 3884\bigr] \)
12.1-d5 \( \bigl[-a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( -159 a^{3} + 252 a^{2} + 801 a - 287\) , \( -2775 a^{3} + 4652 a^{2} + 13093 a - 4702\bigr] \)
12.1-d6 \( \bigl[a\) , \( 2 a^{3} - 3 a^{2} - 11 a\) , \( a^{2} - 2 a - 2\) , \( -323 a^{3} + 504 a^{2} + 1658 a - 609\) , \( -3531 a^{3} + 5452 a^{2} + 18208 a - 6362\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph