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

• Base field: \(\Q(\zeta_{13})^+\)
• Label: 6.6.371293.1-27.1-a
• 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-a over \(\Q(\zeta_{13})^+\)

Isogeny class 27.1-a contains 2 curves linked by isogenies of degree 7.

Curve label	Weierstrass Coefficients
27.1-a1	\( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( -a^{5} + a^{4} + 4 a^{3} - 5 a^{2} - 3 a + 5\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a\) , \( -10 a^{5} + 135 a^{4} + 187 a^{3} - 465 a^{2} - 620 a - 115\) , \( 1078 a^{5} + 2070 a^{4} - 2830 a^{3} - 7811 a^{2} - 4432 a - 668\bigr] \)
27.1-a2	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 6 a + 2\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 4 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( 3 a^{5} - 2 a^{4} - 13 a^{3} + 5 a^{2} + 7 a + 3\) , \( 3 a^{5} - 4 a^{4} - 13 a^{3} + 14 a^{2} + 10 a - 5\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph