Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-256.1-b
Conductor 256.1
Rank \( 0 \)

Related objects

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Base field \(\Q(\sqrt{17}) \)

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

Elliptic curves in class 256.1-b over \(\Q(\sqrt{17}) \)

Isogeny class 256.1-b contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
256.1-b1 \( \bigl[0\) , \( -1\) , \( 0\) , \( -144 a + 360\) , \( -6784 a + 17392\bigr] \)
256.1-b2 \( \bigl[0\) , \( -1\) , \( 0\) , \( -16 a - 24\) , \( -256 a - 400\bigr] \)
256.1-b3 \( \bigl[0\) , \( -1\) , \( 0\) , \( 144 a + 216\) , \( 6784 a + 10608\bigr] \)
256.1-b4 \( \bigl[0\) , \( -1\) , \( 0\) , \( 16 a - 40\) , \( 256 a - 656\bigr] \)
256.1-b5 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15 a - 28\) , \( -16 a - 28\bigr] \)
256.1-b6 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 15697 a - 40284\) , \( -1536816 a + 3936512\bigr] \)
256.1-b7 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 977 a - 2524\) , \( -23856 a + 61184\bigr] \)
256.1-b8 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 17 a - 44\) , \( 0\bigr] \)
256.1-b9 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -175 a - 348\) , \( 1936 a + 2788\bigr] \)
256.1-b10 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -975 a - 1548\) , \( 22880 a + 35780\bigr] \)
256.1-b11 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 177 a - 524\) , \( -2112 a + 5248\bigr] \)
256.1-b12 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15695 a - 24588\) , \( 1521120 a + 2375108\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 24 & 8 & 3 & 6 & 8 & 4 & 12 & 12 & 2 & 24 & 4 \\ 24 & 1 & 3 & 8 & 4 & 12 & 6 & 2 & 8 & 12 & 4 & 24 \\ 8 & 3 & 1 & 24 & 12 & 4 & 2 & 6 & 24 & 4 & 12 & 8 \\ 3 & 8 & 24 & 1 & 2 & 24 & 12 & 4 & 4 & 6 & 8 & 12 \\ 6 & 4 & 12 & 2 & 1 & 12 & 6 & 2 & 2 & 3 & 4 & 6 \\ 8 & 12 & 4 & 24 & 12 & 1 & 2 & 6 & 24 & 4 & 3 & 8 \\ 4 & 6 & 2 & 12 & 6 & 2 & 1 & 3 & 12 & 2 & 6 & 4 \\ 12 & 2 & 6 & 4 & 2 & 6 & 3 & 1 & 4 & 6 & 2 & 12 \\ 12 & 8 & 24 & 4 & 2 & 24 & 12 & 4 & 1 & 6 & 8 & 3 \\ 2 & 12 & 4 & 6 & 3 & 4 & 2 & 6 & 6 & 1 & 12 & 2 \\ 24 & 4 & 12 & 8 & 4 & 3 & 6 & 2 & 8 & 12 & 1 & 24 \\ 4 & 24 & 8 & 12 & 6 & 8 & 4 & 12 & 3 & 2 & 24 & 1 \end{array}\right)\)

Isogeny graph