Elliptic curve isogeny class 15.1-e over number field \(\Q(\sqrt{69}) \)

• Base field: \(\Q(\sqrt{69}) \)
• Label: 2.2.69.1-15.1-e
• Conductor: 15.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 15.1-e over \(\Q(\sqrt{69}) \)

Isogeny class 15.1-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
15.1-e1	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 9 a + 33\) , \( 0\bigr] \)
15.1-e2	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -36 a - 132\) , \( -399 a - 1458\bigr] \)
15.1-e3	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -166 a - 607\) , \( 1600 a + 5845\bigr] \)
15.1-e4	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -626 a - 2297\) , \( -19954 a - 72913\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph