Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.2-b
Conductor 31.2
Rank \( 0 \)

Related objects

Learn more about

Base field \(\Q(\zeta_{15})^+\)

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + 4 x + 1 \); class number \(1\).

Elliptic curves in class 31.2-b over \(\Q(\zeta_{15})^+\)

Isogeny class 31.2-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
31.2-b1 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 416 a^{3} + 76 a^{2} - 1585 a - 348\) , \( 6396 a^{3} + 1587 a^{2} - 23865 a - 5050\bigr] \)
31.2-b2 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( -24 a^{3} - 4 a^{2} + 90 a - 63\) , \( 49 a^{3} + 26 a^{2} - 250 a + 169\bigr] \)
31.2-b3 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 26 a^{3} + 6 a^{2} - 100 a - 23\) , \( 93 a^{3} + 20 a^{2} - 358 a - 75\bigr] \)
31.2-b4 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 36 a^{3} + 16 a^{2} - 135 a - 18\) , \( 78 a^{3} - 11 a^{2} - 319 a - 36\bigr] \)
31.2-b5 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 6 a - 1\bigr] \)
31.2-b6 \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a + 2\) , \( a^{3} - 2 a\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph