Elliptic curve isogeny class 4.1-b over number field \(\Q(\sqrt{57}) \)

• Base field: \(\Q(\sqrt{57}) \)
• Label: 2.2.57.1-4.1-b
• Conductor: 4.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{57}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).

Elliptic curves in class 4.1-b over \(\Q(\sqrt{57}) \)

Isogeny class 4.1-b contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
4.1-b1	\( \bigl[1\) , \( -a\) , \( a\) , \( 91 a - 386\) , \( -817 a + 3486\bigr] \)
4.1-b2	\( \bigl[1\) , \( -a\) , \( a\) , \( a - 1\) , \( -7\bigr] \)
4.1-b3	\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2 a\) , \( -a - 7\bigr] \)
4.1-b4	\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -92 a - 295\) , \( 816 a + 2669\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph