Properties

Base field 4.4.2777.1
Label 4.4.2777.1-512.3-l
Conductor 512.3
Rank \( 1 \)

Related objects

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Base field 4.4.2777.1

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

Elliptic curves in class 512.3-l over 4.4.2777.1

Isogeny class 512.3-l contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
512.3-l1 \( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{2} - a - 2\) , \( -6 a^{3} + 17 a^{2} - 33 a - 34\) , \( -109 a^{3} + 80 a^{2} + 234 a + 88\bigr] \)
512.3-l2 \( \bigl[a^{2} - 2\) , \( -a^{3} + 2 a^{2} + a - 4\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 2 a^{3} - 5 a^{2} - 6 a + 11\) , \( 148 a^{3} - 248 a^{2} - 424 a + 433\bigr] \)
512.3-l3 \( \bigl[a\) , \( -a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -86 a^{3} - 220 a^{2} + 76 a + 115\) , \( 135 a^{3} + 497 a^{2} - 131 a - 265\bigr] \)
512.3-l4 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( -86 a^{3} + 610 a^{2} + 124 a - 2616\) , \( 2513 a^{3} - 12571 a^{2} - 3366 a + 45076\bigr] \)
512.3-l5 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( 104 a^{3} - 25 a^{2} - 431 a - 266\) , \( 902 a^{3} - 185 a^{2} - 3662 a - 2108\bigr] \)
512.3-l6 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( 14 a^{3} - 15 a^{2} - 41 a - 6\) , \( 14 a^{3} - 39 a^{2} + 2 a + 28\bigr] \)
512.3-l7 \( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + 2 a^{2} + a - 4\) , \( a^{2} - a - 2\) , \( 168 a^{3} - 65 a^{2} - 531 a - 335\) , \( -1182 a^{3} - 410 a^{2} + 5982 a + 4093\bigr] \)
512.3-l8 \( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + 2 a^{2} + a - 4\) , \( a^{2} - a - 2\) , \( 33 a^{3} - 90 a^{2} - 11 a + 45\) , \( 247 a^{3} - 606 a^{2} - 45 a + 327\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 12 & 6 & 4 & 12 \\ 3 & 1 & 6 & 12 & 4 & 2 & 12 & 4 \\ 2 & 6 & 1 & 2 & 6 & 3 & 2 & 6 \\ 4 & 12 & 2 & 1 & 3 & 6 & 4 & 12 \\ 12 & 4 & 6 & 3 & 1 & 2 & 12 & 4 \\ 6 & 2 & 3 & 6 & 2 & 1 & 6 & 2 \\ 4 & 12 & 2 & 4 & 12 & 6 & 1 & 3 \\ 12 & 4 & 6 & 12 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph