Properties

Base field \(\Q(\sqrt{13}) \)
Label 2.2.13.1-81.1-a
Conductor 81.1
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 81.1-a over \(\Q(\sqrt{13}) \)

Isogeny class 81.1-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
81.1-a1 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( -4 a + 3\) , \( -4 a + 5\bigr] \)
81.1-a2 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( -43 a - 360\) , \( 1020 a + 3386\bigr] \)
81.1-a3 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( 2381 a - 5352\) , \( -81841 a + 187736\bigr] \)
81.1-a4 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( 2 a\) , \( 3 a + 2\bigr] \)
81.1-a5 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( 2 a - 45\) , \( 39 a - 61\bigr] \)
81.1-a6 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( 131 a - 357\) , \( -1237 a + 2759\bigr] \)
81.1-a7 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( 41 a - 402\) , \( -1021 a + 4406\bigr] \)
81.1-a8 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( 137 a - 585\) , \( 1686 a - 5677\bigr] \)
81.1-a9 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( -133 a - 225\) , \( 1236 a + 1523\bigr] \)
81.1-a10 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( -4 a - 42\) , \( -40 a - 22\bigr] \)
81.1-a11 \( \bigl[a + 1\) , \( -1\) , \( a\) , \( -2383 a - 2970\) , \( 81840 a + 105896\bigr] \)
81.1-a12 \( \bigl[a\) , \( -a\) , \( a + 1\) , \( -139 a - 447\) , \( -1687 a - 3991\bigr] \)

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