Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-25.1-a
Conductor 25.1
Rank \( 0 \)

Related objects

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Base field \(\Q(\sqrt{21}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).

Elliptic curves in class 25.1-a over \(\Q(\sqrt{21}) \)

Isogeny class 25.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
25.1-a1 \( \bigl[a\) , \( a\) , \( a + 1\) , \( -26 a - 43\) , \( 84 a + 148\bigr] \)
25.1-a2 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 66\) , \( -125 a + 369\bigr] \)
25.1-a3 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 71\) , \( -118 a + 338\bigr] \)
25.1-a4 \( \bigl[a\) , \( a\) , \( a + 1\) , \( -a - 3\) , \( -a - 2\bigr] \)
25.1-a5 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 429 a - 1196\) , \( -7403 a + 20663\bigr] \)
25.1-a6 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 4 a - 1\) , \( 9\bigr] \)
25.1-a7 \( \bigl[a\) , \( a\) , \( a + 1\) , \( -26 a - 48\) , \( 72 a + 129\bigr] \)
25.1-a8 \( \bigl[a\) , \( a\) , \( a + 1\) , \( -426 a - 773\) , \( 6632 a + 11894\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 6 & 4 & 12 & 12 & 2 & 4 \\ 3 & 1 & 2 & 12 & 4 & 4 & 6 & 12 \\ 6 & 2 & 1 & 6 & 2 & 2 & 3 & 6 \\ 4 & 12 & 6 & 1 & 12 & 3 & 2 & 4 \\ 12 & 4 & 2 & 12 & 1 & 4 & 6 & 3 \\ 12 & 4 & 2 & 3 & 4 & 1 & 6 & 12 \\ 2 & 6 & 3 & 2 & 6 & 6 & 1 & 2 \\ 4 & 12 & 6 & 4 & 3 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph