Elliptic curve isogeny class 45.1-a over number field 5.5.65657.1

• Base field: 5.5.65657.1
• Label: 5.5.65657.1-45.1-a
• Conductor: 45.1
• Rank: \( 1 \)

Base field 5.5.65657.1

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

Elliptic curves in class 45.1-a over 5.5.65657.1

Isogeny class 45.1-a contains 2 curves linked by isogenies of degree 2.

Curve label	Weierstrass Coefficients
45.1-a1	\( \bigl[-a^{4} + a^{3} + 5 a^{2} - 2 a - 4\) , \( 2 a^{4} - 3 a^{3} - 8 a^{2} + 9 a + 5\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\) , \( -2 a^{4} + 5 a^{3} + 4 a^{2} - 9 a + 1\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 10 a + 5\bigr] \)
45.1-a2	\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 3\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( 16 a^{4} - 20 a^{3} - 76 a^{2} + 58 a + 56\) , \( -11 a^{4} + 10 a^{3} + 56 a^{2} - 13 a - 77\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph