Elliptic curve isogeny class 41.5-b over number field \(\Q(\zeta_{21})^+\)

• Base field: \(\Q(\zeta_{21})^+\)
• Label: 6.6.453789.1-41.5-b
• Conductor: 41.5
• Rank: \( 0 \)

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 41.5-b over \(\Q(\zeta_{21})^+\)

Isogeny class 41.5-b contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
41.5-b1	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\bigr] \)
41.5-b2	\( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 3 a^{2} + 7 a\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( -4 a^{5} + 2 a^{4} + 23 a^{3} - 8 a^{2} - 32 a + 6\) , \( -32 a^{5} + 10 a^{4} + 198 a^{3} - 51 a^{2} - 290 a + 44\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph