Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-676.4-i
Conductor 676.4
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 676.4-i over \(\Q(\sqrt{17}) \)

Isogeny class 676.4-i contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
676.4-i1 \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -47 a - 50\) , \( 189 a + 504\bigr] \)
676.4-i2 \( \bigl[1\) , \( 1\) , \( a\) , \( -25 a + 63\) , \( -16 a + 39\bigr] \)
676.4-i3 \( \bigl[1\) , \( 1\) , \( a\) , \( 345 a - 882\) , \( -5169 a + 13248\bigr] \)
676.4-i4 \( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 3 a + 5\) , \( a - 7\bigr] \)
676.4-i5 \( \bigl[1\) , \( 1\) , \( a\) , \( -319 a - 497\) , \( 124 a + 193\bigr] \)
676.4-i6 \( \bigl[1\) , \( 0\) , \( a + 1\) , \( 199 a - 1990\) , \( -14489 a + 19668\bigr] \)
676.4-i7 \( \bigl[1\) , \( 0\) , \( a + 1\) , \( -81 a - 270\) , \( 743 a + 1812\bigr] \)
676.4-i8 \( \bigl[1\) , \( 0\) , \( a + 1\) , \( -21 a - 35\) , \( -78 a - 122\bigr] \)
676.4-i9 \( \bigl[1\) , \( a\) , \( 1\) , \( 2806 a - 7189\) , \( 114963 a - 294484\bigr] \)
676.4-i10 \( \bigl[1\) , \( a\) , \( 1\) , \( 956 a - 2464\) , \( -22281 a + 57086\bigr] \)
676.4-i11 \( \bigl[1\) , \( 0\) , \( a + 1\) , \( -331 a - 545\) , \( -4886 a - 7574\bigr] \)
676.4-i12 \( \bigl[1\) , \( a\) , \( 1\) , \( 2916 a - 7744\) , \( 99831 a - 254210\bigr] \)

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