Elliptic curve isogeny class 12.2-b over number field \(\Q(\sqrt{10}) \)

• Base field: \(\Q(\sqrt{10}) \)
• Label: 2.2.40.1-12.2-b
• Conductor: 12.2
• Rank: \( 1 \)

Base field \(\Q(\sqrt{10}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).

Elliptic curves in class 12.2-b over \(\Q(\sqrt{10}) \)

Isogeny class 12.2-b contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
12.2-b1	\( \bigl[a\) , \( a\) , \( a\) , \( 135 a - 421\) , \( -1518 a + 4805\bigr] \)
12.2-b2	\( \bigl[a\) , \( a\) , \( a\) , \( 10 a - 26\) , \( -24 a + 80\bigr] \)
12.2-b3	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 14 a - 39\) , \( -a + 5\bigr] \)
12.2-b4	\( \bigl[a\) , \( a\) , \( a\) , \( 5 a - 11\) , \( -50 a + 161\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph