Elliptic curve isogeny class 16.5-b over number field \(\Q(\sqrt{41}) \)

• Base field: \(\Q(\sqrt{41}) \)
• Label: 2.2.41.1-16.5-b
• Conductor: 16.5
• Rank: \( 1 \)

Base field \(\Q(\sqrt{41}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 10 \); class number \(1\).

Elliptic curves in class 16.5-b over \(\Q(\sqrt{41}) \)

Isogeny class 16.5-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
16.5-b1	\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 6 a + 6\) , \( a + 25\bigr] \)
16.5-b2	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 18 a + 51\) , \( 16 a + 40\bigr] \)
16.5-b3	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -7 a - 14\) , \( -6 a - 14\bigr] \)
16.5-b4	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 323 a - 1204\) , \( 324 a - 1204\bigr] \)
16.5-b5	\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -11\) , \( 12 a - 50\bigr] \)
16.5-b6	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -1302 a + 4811\) , \( -3154 a + 11670\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 8 & 4 \\ 8 & 1 & 2 & 4 & 4 & 8 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 2 & 4 & 2 & 1 & 4 & 2 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 8 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph