Elliptic curve isogeny class 57.1-g over number field \(\Q(\sqrt{57}) \)

• Base field: \(\Q(\sqrt{57}) \)
• Label: 2.2.57.1-57.1-g
• Conductor: 57.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 57.1-g over \(\Q(\sqrt{57}) \)

Isogeny class 57.1-g contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
57.1-g1	\( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \)
57.1-g2	\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 543 a - 2315\) , \( 13825 a - 59110\bigr] \)
57.1-g3	\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \)
57.1-g4	\( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \)
57.1-g5	\( \bigl[1\) , \( a + 1\) , \( a\) , \( -542 a - 1772\) , \( -14368 a - 47057\bigr] \)
57.1-g6	\( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \)

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