Properties

Base field 4.4.2777.1
Label 4.4.2777.1-64.3-b
Conductor 64.3
Rank \( 0 \)

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 64.3-b over 4.4.2777.1

Isogeny class 64.3-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
64.3-b1 \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{2} - 2\) , \( 29 a^{3} - 76 a^{2} + 35\) , \( -41 a^{3} + 101 a^{2} + 12 a - 56\bigr] \)
64.3-b2 \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{2} - 2\) , \( 399 a^{3} - 1011 a^{2} - 55 a + 505\) , \( -8414 a^{3} + 21138 a^{2} + 1713 a - 11090\bigr] \)
64.3-b3 \( \bigl[a^{2} - a - 2\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -2 a^{3} + 2 a^{2} + 6 a - 1\) , \( -5 a^{3} + 7 a^{2} + 14 a - 12\bigr] \)
64.3-b4 \( \bigl[a^{2} - 2\) , \( a^{2} - 2 a - 2\) , \( 0\) , \( 3 a^{3} - 5 a^{2} - 6 a - 1\) , \( 2 a^{3} - 15 a^{2} + 14 a + 17\bigr] \)
64.3-b5 \( \bigl[a^{2} - 2\) , \( a^{2} - 2 a - 2\) , \( 0\) , \( 38 a^{3} - 90 a^{2} - 16 a + 39\) , \( 273 a^{3} - 691 a^{2} - 51 a + 367\bigr] \)
64.3-b6 \( \bigl[a^{3} - 4 a\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -33 a^{3} - 40 a^{2} + 22 a + 24\) , \( -293 a^{3} - 393 a^{2} + 229 a + 245\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph