Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-28.1-a
Conductor 28.1
Rank \( 0 \)

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 28.1-a over \(\Q(\sqrt{2}) \)

Isogeny class 28.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
28.1-a1 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -9 a - 18\) , \( -320 a - 467\bigr] \)
28.1-a2 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( a + 2\) , \( 12 a + 17\bigr] \)
28.1-a3 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -19 a - 148\) , \( 318 a - 201\bigr] \)
28.1-a4 \( \bigl[0\) , \( -1\) , \( 0\) , \( -2 a + 3\) , \( 0\bigr] \)
28.1-a5 \( \bigl[a\) , \( -1\) , \( a\) , \( 2 a - 4\) , \( -a + 1\bigr] \)
28.1-a6 \( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -34 a - 53\) , \( -133 a - 203\bigr] \)
28.1-a7 \( \bigl[a\) , \( -1\) , \( a\) , \( 17 a - 29\) , \( 49 a - 66\bigr] \)
28.1-a8 \( \bigl[0\) , \( -1\) , \( 0\) , \( 18 a - 37\) , \( 68 a - 108\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph