Elliptic curve isogeny class 127.2-a over number field \(\Q(\zeta_{7})^+\)

• Base field: \(\Q(\zeta_{7})^+\)
• Label: 3.3.49.1-127.2-a
• Conductor: 127.2
• Rank: \( 0 \)

Base field \(\Q(\zeta_{7})^+\)

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 2 x + 1 \); class number \(1\).

Elliptic curves in class 127.2-a over \(\Q(\zeta_{7})^+\)

Isogeny class 127.2-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
127.2-a1	\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 3\) , \( a^{2} - 1\) , \( -5 a^{2} + 15 a - 54\) , \( -121 a^{2} + 111 a + 20\bigr] \)
127.2-a2	\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 3\) , \( a^{2} - 1\) , \( 20 a^{2} - 10 a - 44\) , \( -31 a^{2} + 18 a + 70\bigr] \)
127.2-a3	\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 3\) , \( a^{2} - 1\) , \( -5 a^{2} - 10 a - 9\) , \( -21 a^{2} - 43 a - 27\bigr] \)
127.2-a4	\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 3\) , \( a^{2} - 1\) , \( 1\) , \( 0\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph