Elliptic curve isogeny class 27.1-b over number field \(\Q(\zeta_{13})^+\)

• Base field: \(\Q(\zeta_{13})^+\)
• Label: 6.6.371293.1-27.1-b
• Conductor: 27.1
• Rank: \( 0 \)

Base field \(\Q(\zeta_{13})^+\)

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

Elliptic curves in class 27.1-b over \(\Q(\zeta_{13})^+\)

Isogeny class 27.1-b contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
27.1-b1	\( \bigl[a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( a^{5} - 4 a^{3} - a^{2} + a + 3\) , \( a^{4} - 4 a^{2} + 3\) , \( -a^{5} + a^{4} + 2 a^{3} - 3 a^{2} - a + 8\) , \( a^{4} - 4 a^{3} - 2 a^{2} + 6 a + 2\bigr] \)
27.1-b2	\( \bigl[a^{5} - 4 a^{3} + a^{2} + 3 a - 1\) , \( a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 2 a - 5\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 3 a + 3\) , \( 91 a^{5} + 53 a^{4} - 378 a^{3} - 238 a^{2} + 196 a + 60\) , \( 788 a^{5} + 381 a^{4} - 3365 a^{3} - 1823 a^{2} + 1998 a + 545\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph