Elliptic curve isogeny class 44.4-a over number field \(\Q(\sqrt{-7}) \)

• Base field: \(\Q(\sqrt{-7}) \)
• Label: 2.0.7.1-44.4-a
• Conductor: 44.4
• Rank: \( 0 \)

Base field \(\Q(\sqrt{-7}) \)

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

Elliptic curves in class 44.4-a over \(\Q(\sqrt{-7}) \)

Isogeny class 44.4-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
44.4-a1	\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 55 a + 36\) , \( -87 a + 446\bigr] \)
44.4-a2	\( \bigl[1\) , \( 1\) , \( a\) , \( 3 a + 37\) , \( -136 a + 219\bigr] \)
44.4-a3	\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 1\) , \( 1\bigr] \)
44.4-a4	\( \bigl[1\) , \( a\) , \( 1\) , \( -14 a - 3\) , \( 19 a - 14\bigr] \)
44.4-a5	\( \bigl[1\) , \( 1\) , \( a\) , \( -2 a + 2\) , \( -a + 2\bigr] \)
44.4-a6	\( \bigl[1\) , \( 1\) , \( a\) , \( 13 a - 23\) , \( -28 a + 19\bigr] \)
44.4-a7	\( \bigl[1\) , \( a\) , \( 1\) , \( a + 2\) , \( 3 a - 2\bigr] \)
44.4-a8	\( \bigl[1\) , \( 1\) , \( a\) , \( -22 a + 22\) , \( -21 a + 78\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph