Elliptic curve isogeny class 4.1-a over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-4.1-a
• Conductor: 4.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{17}) \)

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

Elliptic curves in class 4.1-a over \(\Q(\sqrt{17}) \)

Isogeny class 4.1-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label	Weierstrass Coefficients
4.1-a1	\( \bigl[1\) , \( 0\) , \( 1\) , \( -9 a + 22\) , \( 106 a - 272\bigr] \)
4.1-a2	\( \bigl[1\) , \( 0\) , \( 1\) , \( -a - 2\) , \( 4 a + 6\bigr] \)
4.1-a3	\( \bigl[1\) , \( 0\) , \( 1\) , \( 9 a + 13\) , \( -106 a - 166\bigr] \)
4.1-a4	\( \bigl[1\) , \( 0\) , \( 1\) , \( a - 3\) , \( -4 a + 10\bigr] \)
4.1-a5	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -2 a\) , \( 0\bigr] \)
4.1-a6	\( \bigl[1\) , \( a\) , \( a + 1\) , \( 981 a - 2517\) , \( 23628 a - 60528\bigr] \)
4.1-a7	\( \bigl[1\) , \( a\) , \( a + 1\) , \( 61 a - 157\) , \( 348 a - 896\bigr] \)
4.1-a8	\( \bigl[1\) , \( a\) , \( a + 1\) , \( a - 2\) , \( -a\bigr] \)
4.1-a9	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -12 a - 20\) , \( -28 a - 44\bigr] \)
4.1-a10	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -62 a - 95\) , \( -349 a - 547\bigr] \)
4.1-a11	\( \bigl[1\) , \( a\) , \( a + 1\) , \( 11 a - 32\) , \( 27 a - 72\bigr] \)
4.1-a12	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -982 a - 1535\) , \( -23629 a - 36899\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 24 & 8 & 3 & 6 & 8 & 4 & 12 & 12 & 2 & 24 & 4 \\ 24 & 1 & 3 & 8 & 4 & 12 & 6 & 2 & 8 & 12 & 4 & 24 \\ 8 & 3 & 1 & 24 & 12 & 4 & 2 & 6 & 24 & 4 & 12 & 8 \\ 3 & 8 & 24 & 1 & 2 & 24 & 12 & 4 & 4 & 6 & 8 & 12 \\ 6 & 4 & 12 & 2 & 1 & 12 & 6 & 2 & 2 & 3 & 4 & 6 \\ 8 & 12 & 4 & 24 & 12 & 1 & 2 & 6 & 24 & 4 & 3 & 8 \\ 4 & 6 & 2 & 12 & 6 & 2 & 1 & 3 & 12 & 2 & 6 & 4 \\ 12 & 2 & 6 & 4 & 2 & 6 & 3 & 1 & 4 & 6 & 2 & 12 \\ 12 & 8 & 24 & 4 & 2 & 24 & 12 & 4 & 1 & 6 & 8 & 3 \\ 2 & 12 & 4 & 6 & 3 & 4 & 2 & 6 & 6 & 1 & 12 & 2 \\ 24 & 4 & 12 & 8 & 4 & 3 & 6 & 2 & 8 & 12 & 1 & 24 \\ 4 & 24 & 8 & 12 & 6 & 8 & 4 & 12 & 3 & 2 & 24 & 1 \end{array}\right)\)

Isogeny graph