Elliptic curve isogeny class 256.1-r over number field \(\Q(\sqrt{57}) \)

• Base field: \(\Q(\sqrt{57}) \)
• Label: 2.2.57.1-256.1-r
• Conductor: 256.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 256.1-r over \(\Q(\sqrt{57}) \)

Isogeny class 256.1-r contains 4 curves linked by isogenies of degrees dividing 21.

Curve label	Weierstrass Coefficients
256.1-r1	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 1142337 a - 4884238\) , \( 1290582430 a - 5517144100\bigr] \)
256.1-r2	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1142335 a - 3741902\) , \( -1289440094 a - 4222819768\bigr] \)
256.1-r3	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -3423 a + 14642\) , \( -190018 a + 812316\bigr] \)
256.1-r4	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 3425 a + 11218\) , \( 186594 a + 611080\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)

Isogeny graph