Elliptic curve isogeny class 225.1-d over number field \(\Q(\sqrt{21}) \)

• Base field: \(\Q(\sqrt{21}) \)
• Label: 2.2.21.1-225.1-d
• Conductor: 225.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 225.1-d over \(\Q(\sqrt{21}) \)

Isogeny class 225.1-d contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
225.1-d1	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -287 a + 794\) , \( -1612 a + 4500\bigr] \)
225.1-d2	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 28 a - 23\) , \( 42 a - 188\bigr] \)
225.1-d3	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 13 a - 53\) , \( 51 a - 149\bigr] \)
225.1-d4	\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -25 a - 46\) , \( 59 a + 105\bigr] \)
225.1-d5	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 238 a - 803\) , \( 3696 a - 9674\bigr] \)
225.1-d6	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -2 a - 8\) , \( -3 a - 8\bigr] \)
225.1-d7	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 73 a - 211\) , \( -73 a + 204\bigr] \)
225.1-d8	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 673 a - 1936\) , \( 15542 a - 43236\bigr] \)

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