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Properties

Base field \(\Q(\sqrt{13}) \)
Label 2.2.13.1-9.1-a
Conductor 9.1
Rank \( 0 \)

Related objects

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Base field \(\Q(\sqrt{13}) \)

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

Elliptic curves in class 9.1-a over \(\Q(\sqrt{13}) \)

Isogeny class 9.1-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
9.1-a1 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \)
9.1-a2 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5 a - 40\) , \( -56 a - 157\bigr] \)
9.1-a3 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 265 a - 594\) , \( 3141 a - 7218\bigr] \)
9.1-a4 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 0\) , \( 0\bigr] \)
9.1-a5 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5\) , \( -3 a - 1\bigr] \)
9.1-a6 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 15 a - 39\) , \( 54 a - 117\bigr] \)
9.1-a7 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a - 44\) , \( 51 a - 168\bigr] \)
9.1-a8 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 15 a - 65\) , \( -69 a + 182\bigr] \)
9.1-a9 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -15 a - 25\) , \( -69 a - 88\bigr] \)
9.1-a10 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( 3 a + 1\bigr] \)
9.1-a11 \( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -265 a - 330\) , \( -3406 a - 4407\bigr] \)
9.1-a12 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -15 a - 49\) , \( 84 a + 163\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 24 & 8 & 3 & 6 & 4 & 8 & 12 & 12 & 2 & 24 & 4 \\ 24 & 1 & 3 & 8 & 4 & 6 & 12 & 8 & 2 & 12 & 4 & 24 \\ 8 & 3 & 1 & 24 & 12 & 2 & 4 & 24 & 6 & 4 & 12 & 8 \\ 3 & 8 & 24 & 1 & 2 & 12 & 24 & 4 & 4 & 6 & 8 & 12 \\ 6 & 4 & 12 & 2 & 1 & 6 & 12 & 2 & 2 & 3 & 4 & 6 \\ 4 & 6 & 2 & 12 & 6 & 1 & 2 & 12 & 3 & 2 & 6 & 4 \\ 8 & 12 & 4 & 24 & 12 & 2 & 1 & 24 & 6 & 4 & 3 & 8 \\ 12 & 8 & 24 & 4 & 2 & 12 & 24 & 1 & 4 & 6 & 8 & 3 \\ 12 & 2 & 6 & 4 & 2 & 3 & 6 & 4 & 1 & 6 & 2 & 12 \\ 2 & 12 & 4 & 6 & 3 & 2 & 4 & 6 & 6 & 1 & 12 & 2 \\ 24 & 4 & 12 & 8 & 4 & 6 & 3 & 8 & 2 & 12 & 1 & 24 \\ 4 & 24 & 8 & 12 & 6 & 4 & 8 & 3 & 12 & 2 & 24 & 1 \end{array}\right)\)

Isogeny graph