Elliptic curve isogeny class 25.1-b over number field \(\Q(\sqrt{21}) \)

• Base field: \(\Q(\sqrt{21}) \)
• Label: 2.2.21.1-25.1-b
• Conductor: 25.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{21}) \)

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

Elliptic curves in class 25.1-b over \(\Q(\sqrt{21}) \)

Isogeny class 25.1-b contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
25.1-b1	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -18 a + 58\) , \( -20 a + 51\bigr] \)
25.1-b2	\( \bigl[a\) , \( a\) , \( a\) , \( 22 a + 33\) , \( 56 a + 96\bigr] \)
25.1-b3	\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 12\) , \( -4 a - 11\bigr] \)
25.1-b4	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a - 7\) , \( -5 a + 26\bigr] \)
25.1-b5	\( \bigl[a\) , \( a\) , \( a\) , \( -28 a - 137\) , \( -284 a - 786\bigr] \)
25.1-b6	\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 7\) , \( a + 1\bigr] \)
25.1-b7	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a - 12\) , \( -5 a + 9\bigr] \)
25.1-b8	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 32 a - 162\) , \( 150 a - 921\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 6 & 4 & 12 & 12 & 2 & 4 \\ 3 & 1 & 2 & 12 & 4 & 4 & 6 & 12 \\ 6 & 2 & 1 & 6 & 2 & 2 & 3 & 6 \\ 4 & 12 & 6 & 1 & 12 & 3 & 2 & 4 \\ 12 & 4 & 2 & 12 & 1 & 4 & 6 & 3 \\ 12 & 4 & 2 & 3 & 4 & 1 & 6 & 12 \\ 2 & 6 & 3 & 2 & 6 & 6 & 1 & 2 \\ 4 & 12 & 6 & 4 & 3 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph