Elliptic curve isogeny class 41.1-c over number field \(\Q(\zeta_{21})^+\)

• Base field: \(\Q(\zeta_{21})^+\)
• Label: 6.6.453789.1-41.1-c
• Conductor: 41.1
• 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 41.1-c over \(\Q(\zeta_{21})^+\)

Isogeny class 41.1-c contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
41.1-c1	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( -a^{5} + 5 a^{3} - 2 a^{2} - 4 a + 5\) , \( a^{5} - 5 a^{3} + a^{2} + 6 a - 1\) , \( -3 a^{5} - 2 a^{4} + 13 a^{3} + 5 a^{2} - 12 a + 4\) , \( -a^{5} - a^{4} + 5 a^{3} + 2 a^{2} - 5 a + 1\bigr] \)
41.1-c2	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 4 a - 3\) , \( a^{3} - 3 a\) , \( 8 a^{5} + 3 a^{4} - 45 a^{3} - 2 a^{2} + 62 a - 8\) , \( -4 a^{5} + 9 a^{4} + 37 a^{3} - 30 a^{2} - 71 a + 11\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph