Elliptic curve isogeny class 32.1-a over number field \(\Q(\zeta_{16})^+\)

• Base field: \(\Q(\zeta_{16})^+\)
• Label: 4.4.2048.1-32.1-a
• Conductor: 32.1
• Rank: \( 0 \)

Base field \(\Q(\zeta_{16})^+\)

Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 2 \); class number \(1\).

Elliptic curves in class 32.1-a over \(\Q(\zeta_{16})^+\)

Isogeny class 32.1-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label	Weierstrass Coefficients
32.1-a1	\( \bigl[a^{2} - 2\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \)
32.1-a2	\( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 2\) , \( -3\) , \( 0\bigr] \)
32.1-a3	\( \bigl[a^{3} - 3 a\) , \( a^{2} - 1\) , \( a^{3} - 2 a\) , \( a^{2} - 6\) , \( -6 a^{2} + 19\bigr] \)
32.1-a4	\( \bigl[a^{3} - 3 a\) , \( a^{2} - 1\) , \( a\) , \( -3\) , \( -6 a^{2}\bigr] \)
32.1-a5	\( \bigl[a^{3} - 2 a\) , \( a^{2} - 3\) , \( 0\) , \( -a^{2} + 5\) , \( a^{2} - 3\bigr] \)
32.1-a6	\( \bigl[a^{3} - 2 a\) , \( a^{2} - 3\) , \( a^{3} - 2 a\) , \( -2\) , \( -a^{2} + 2\bigr] \)
32.1-a7	\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - 3 a\) , \( -a^{2} - 1\) , \( 6 a^{2} - 24\bigr] \)
32.1-a8	\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - 2 a\) , \( -2 a^{2}\) , \( 5 a^{2} - 3\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 8 & 2 & 4 & 2 & 2 & 8 \\ 4 & 1 & 2 & 8 & 4 & 2 & 8 & 2 \\ 8 & 2 & 1 & 16 & 8 & 4 & 16 & 4 \\ 2 & 8 & 16 & 1 & 8 & 4 & 4 & 16 \\ 4 & 4 & 8 & 8 & 1 & 2 & 8 & 8 \\ 2 & 2 & 4 & 4 & 2 & 1 & 4 & 4 \\ 2 & 8 & 16 & 4 & 8 & 4 & 1 & 16 \\ 8 & 2 & 4 & 16 & 8 & 4 & 16 & 1 \end{array}\right)\)

Isogeny graph