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Properties

Base field \(\Q(\sqrt{13}) \)
Label 2.2.13.1-27.2-a
Conductor 27.2
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 27.2-a over \(\Q(\sqrt{13}) \)

Isogeny class 27.2-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
27.2-a1 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a + 1\) , \( -2 a + 2\bigr] \)
27.2-a2 \( \bigl[1\) , \( -a\) , \( 1\) , \( 26 a - 145\) , \( -340 a + 350\bigr] \)
27.2-a3 \( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 849 a + 1064\) , \( 21278 a + 27823\bigr] \)
27.2-a4 \( \bigl[1\) , \( -a\) , \( 1\) , \( a\) , \( -2\bigr] \)
27.2-a5 \( \bigl[1\) , \( -a\) , \( 1\) , \( 6 a - 20\) , \( 20 a - 46\bigr] \)
27.2-a6 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 82 a - 204\) , \( -547 a + 1264\bigr] \)
27.2-a7 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 57 a - 194\) , \( -767 a + 1595\bigr] \)
27.2-a8 \( \bigl[1\) , \( -a\) , \( 1\) , \( 111 a - 305\) , \( 1130 a - 2434\bigr] \)
27.2-a9 \( \bigl[1\) , \( -a\) , \( 1\) , \( -19 a - 55\) , \( -70 a - 154\bigr] \)
27.2-a10 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2 a - 19\) , \( -2 a + 11\bigr] \)
27.2-a11 \( \bigl[1\) , \( -a\) , \( 1\) , \( -464 a - 525\) , \( -6420 a - 8630\bigr] \)
27.2-a12 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2 a - 154\) , \( 403 a - 286\bigr] \)

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