Elliptic curve isogeny class 171.1-f over number field \(\Q(\sqrt{19}) \)

• Base field: \(\Q(\sqrt{19}) \)
• Label: 2.2.76.1-171.1-f
• Conductor: 171.1
• Rank: \( 1 \)

Base field \(\Q(\sqrt{19}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).

Elliptic curves in class 171.1-f over \(\Q(\sqrt{19}) \)

Isogeny class 171.1-f contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
171.1-f1	\( \bigl[a\) , \( 1\) , \( a\) , \( 175 a - 860\) , \( 3178 a - 14456\bigr] \)
171.1-f2	\( \bigl[a\) , \( 1\) , \( a\) , \( 10\) , \( -10\bigr] \)
171.1-f3	\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \)
171.1-f4	\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -16\bigr] \)
171.1-f5	\( \bigl[a\) , \( 1\) , \( a\) , \( -100\) , \( -586\bigr] \)
171.1-f6	\( \bigl[a\) , \( 1\) , \( a\) , \( -175 a - 860\) , \( -3178 a - 14456\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 8 & 4 & 2 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 8 & 4 & 1 & 2 & 4 & 8 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 2 & 4 & 4 & 2 & 1 & 2 \\ 4 & 8 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph