Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-676.5-i
Conductor 676.5
Rank \( 1 \)

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 676.5-i over \(\Q(\sqrt{17}) \)

Isogeny class 676.5-i contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
676.5-i1 \( \bigl[1\) , \( 1\) , \( a + 1\) , \( -346 a - 537\) , \( 5168 a + 8079\bigr] \)
676.5-i2 \( \bigl[1\) , \( a\) , \( 0\) , \( -3 a + 8\) , \( -a - 6\bigr] \)
676.5-i3 \( \bigl[1\) , \( a\) , \( 0\) , \( 47 a - 97\) , \( -189 a + 693\bigr] \)
676.5-i4 \( \bigl[1\) , \( 1\) , \( a + 1\) , \( 24 a + 38\) , \( 15 a + 23\bigr] \)
676.5-i5 \( \bigl[1\) , \( 0\) , \( a\) , \( 20 a - 55\) , \( 77 a - 199\bigr] \)
676.5-i6 \( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -2916 a - 4828\) , \( -99831 a - 154379\bigr] \)
676.5-i7 \( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -956 a - 1508\) , \( 22281 a + 34805\bigr] \)
676.5-i8 \( \bigl[1\) , \( 1\) , \( a + 1\) , \( 318 a - 816\) , \( -125 a + 317\bigr] \)
676.5-i9 \( \bigl[1\) , \( 0\) , \( a\) , \( 330 a - 875\) , \( 4885 a - 12459\bigr] \)
676.5-i10 \( \bigl[1\) , \( 0\) , \( a\) , \( 80 a - 350\) , \( -744 a + 2556\bigr] \)
676.5-i11 \( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -2806 a - 4383\) , \( -114963 a - 179521\bigr] \)
676.5-i12 \( \bigl[1\) , \( 0\) , \( a\) , \( -200 a - 1790\) , \( 14488 a + 5180\bigr] \)

Rank

Rank: \( 1 \)

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