Elliptic curve isogeny class 53.1-a over number field \(\Q(\zeta_{13})^+\)

• Base field: \(\Q(\zeta_{13})^+\)
• Label: 6.6.371293.1-53.1-a
• Conductor: 53.1
• Rank: \( 1 \)

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 53.1-a over \(\Q(\zeta_{13})^+\)

Isogeny class 53.1-a contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
53.1-a1	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 3 a^{5} - 3 a^{4} - 17 a^{3} + 7 a^{2} + 23 a + 5\) , \( a^{5} - a^{4} - 7 a^{3} + 2 a^{2} + 11 a + 2\bigr] \)
53.1-a2	\( \bigl[a^{2} + a - 1\) , \( a^{5} - a^{4} - 6 a^{3} + 3 a^{2} + 9 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( a^{5} - 2 a^{4} - 8 a^{3} + 6 a^{2} + 17 a + 4\) , \( 2 a^{5} - a^{4} - 11 a^{3} + 2 a^{2} + 15 a + 4\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph