Properties

Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Label 4.4.2304.1-47.2-a
Conductor 47.2
Rank \( 0 \)

Related objects

Learn more

Base field \(\Q(\sqrt{2}, \sqrt{3})\)

Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 1 \); class number \(1\).

Elliptic curves in class 47.2-a over \(\Q(\sqrt{2}, \sqrt{3})\)

Isogeny class 47.2-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
47.2-a1 \( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 3 a^{3} - 2 a^{2} - 15 a - 6\) , \( -7 a^{3} - 5 a^{2} + 15 a + 8\bigr] \)
47.2-a2 \( \bigl[a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 1\) , \( a\) , \( -2 a^{3} + 2 a^{2} + 3 a\) , \( -a^{3} + a^{2} + 2 a - 1\bigr] \)
47.2-a3 \( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{2} - 2\) , \( 8 a^{3} + 5 a^{2} - 29 a - 15\) , \( 16 a^{3} + 9 a^{2} - 60 a - 32\bigr] \)
47.2-a4 \( \bigl[a^{2} - 1\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a\) , \( a^{2} + 3 a\) , \( a^{3}\bigr] \)

Rank

Rank: \( 0 \)

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