Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-1024.1-k
Conductor 1024.1
Rank \( 1 \)

Related objects

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Base field \(\Q(\sqrt{2}) \)

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

Elliptic curves in class 1024.1-k over \(\Q(\sqrt{2}) \)

Isogeny class 1024.1-k contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
1024.1-k1 \( \bigl[0\) , \( 0\) , \( 0\) , \( 120 a - 182\) , \( 924 a - 1232\bigr] \)
1024.1-k2 \( \bigl[0\) , \( 0\) , \( 0\) , \( 120 a - 182\) , \( -924 a + 1232\bigr] \)
1024.1-k3 \( \bigl[0\) , \( 0\) , \( 0\) , \( 2\) , \( 0\bigr] \)
1024.1-k4 \( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \)
1024.1-k5 \( \bigl[0\) , \( 0\) , \( 0\) , \( -22\) , \( -28 a\bigr] \)
1024.1-k6 \( \bigl[0\) , \( 0\) , \( 0\) , \( -22\) , \( 28 a\bigr] \)
1024.1-k7 \( \bigl[0\) , \( 0\) , \( 0\) , \( -120 a - 182\) , \( -924 a - 1232\bigr] \)
1024.1-k8 \( \bigl[0\) , \( 0\) , \( 0\) , \( -120 a - 182\) , \( 924 a + 1232\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph