Elliptic curve isogeny class 81.1-e over number field \(\Q(\sqrt{69}) \)

• Base field: \(\Q(\sqrt{69}) \)
• Label: 2.2.69.1-81.1-e
• Conductor: 81.1
• Rank: \( 1 \)

Base field \(\Q(\sqrt{69}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 17 \); class number \(1\).

Elliptic curves in class 81.1-e over \(\Q(\sqrt{69}) \)

Isogeny class 81.1-e contains 4 curves linked by isogenies of degrees dividing 27.

Curve label	Weierstrass Coefficients
81.1-e1	\( \bigl[0\) , \( 0\) , \( 1\) , \( -2250 a - 8220\) , \( -118404 a - 432567\bigr] \)
81.1-e2	\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \)
81.1-e3	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 468 a - 2178\bigr] \)
81.1-e4	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 27 & 3 & 9 \\ 27 & 1 & 9 & 3 \\ 3 & 9 & 1 & 3 \\ 9 & 3 & 3 & 1 \end{array}\right)\)

Isogeny graph