Elliptic curve isogeny class 41.4-c over number field \(\Q(\zeta_{21})^+\)

• Base field: \(\Q(\zeta_{21})^+\)
• Label: 6.6.453789.1-41.4-c
• Conductor: 41.4
• 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 41.4-c over \(\Q(\zeta_{21})^+\)

Isogeny class 41.4-c contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
41.4-c1	\( \bigl[a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - a + 2\) , \( a^{5} - 4 a^{3} + 2 a^{2} + 2 a - 4\) , \( 3 a^{5} + 2 a^{4} - 11 a^{3} - 4 a^{2} + 8 a + 1\) , \( 6 a^{5} + 6 a^{4} - 24 a^{3} - 14 a^{2} + 19 a - 2\bigr] \)
41.4-c2	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 1\) , \( -3 a^{5} - a^{4} + 18 a^{3} + 5 a^{2} - 20 a + 3\) , \( 17 a^{5} + 11 a^{4} - 82 a^{3} - 31 a^{2} + 79 a - 11\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph