Elliptic curve isogeny class 41.2-c over number field \(\Q(\zeta_{21})^+\)

• Base field: \(\Q(\zeta_{21})^+\)
• Label: 6.6.453789.1-41.2-c
• Conductor: 41.2
• Rank: \( 1 \)

Base field \(\Q(\zeta_{21})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).

Elliptic curves in class 41.2-c over \(\Q(\zeta_{21})^+\)

Isogeny class 41.2-c contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
41.2-c1	\( \bigl[a^{4} - 3 a^{2} + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 2 a^{2} - a + 1\) , \( a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 9 a + 1\) , \( 3 a^{5} - a^{4} - 20 a^{3} + 5 a^{2} + 31 a - 5\bigr] \)
41.2-c2	\( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 4\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( -2 a^{4} - a^{3} + 8 a^{2} + a - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph