Properties

Base field 4.4.2777.1
Label 4.4.2777.1-64.3-c
Conductor 64.3
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 64.3-c over 4.4.2777.1

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

Curve label Weierstrass Coefficients
64.3-c1 \( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{2} - a - 2\) , \( -7 a^{3} - 14 a^{2} - 5 a - 26\) , \( 18 a^{3} + 60 a^{2} + 82 a - 16\bigr] \)
64.3-c2 \( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{2} - a - 2\) , \( -12 a^{3} - 14 a^{2} + 10 a + 4\) , \( 42 a^{3} + 54 a^{2} - 34 a - 32\bigr] \)
64.3-c3 \( \bigl[a\) , \( 0\) , \( a^{3} - 4 a\) , \( 8 a^{3} - 32 a^{2} - 39 a + 45\) , \( 76 a^{3} - 57 a^{2} - 167 a + 137\bigr] \)
64.3-c4 \( \bigl[a\) , \( -a^{2} + 2\) , \( a\) , \( -a^{2} + 1\) , \( 0\bigr] \)
64.3-c5 \( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -43 a^{3} + 70 a^{2} + 131 a - 144\) , \( 244 a^{3} - 402 a^{2} - 721 a + 725\bigr] \)
64.3-c6 \( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -3 a^{3} + 5 a^{2} + 6 a - 14\) , \( 2 a^{3} - 6 a^{2} - 5 a + 13\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph