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

• Base field: \(\Q(\sqrt{57}) \)
• Label: 2.2.57.1-4.1-a
• 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-a over \(\Q(\sqrt{57}) \)

Isogeny class 4.1-a contains 4 curves linked by isogenies of degrees dividing 21.

Curve label	Weierstrass Coefficients
4.1-a1	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 71395 a - 305260\) , \( -20071186 a + 85802856\bigr] \)
4.1-a2	\( \bigl[1\) , \( a\) , \( a + 1\) , \( -71396 a - 233865\) , \( 20071185 a + 65731670\bigr] \)
4.1-a3	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -215 a + 920\) , \( 2686 a - 11488\bigr] \)
4.1-a4	\( \bigl[1\) , \( a\) , \( a + 1\) , \( 214 a + 705\) , \( -2687 a - 8802\bigr] \)

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