Properties

Base field \(\Q(\sqrt{13}) \)
Label 2.2.13.1-4.1-a
Conductor 4.1
Rank \( 0 \)

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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 4.1-a over \(\Q(\sqrt{13}) \)

Isogeny class 4.1-a contains 6 curves linked by isogenies of degrees dividing 45.

Curve label Weierstrass Coefficients
4.1-a1 \( \bigl[1\) , \( 1\) , \( a\) , \( -29 a + 2\) , \( -52 a - 106\bigr] \)
4.1-a2 \( \bigl[1\) , \( 1\) , \( a + 1\) , \( -2 a - 2\) , \( 0\bigr] \)
4.1-a3 \( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 75 a - 172\) , \( 507 a - 1170\bigr] \)
4.1-a4 \( \bigl[1\) , \( 1\) , \( a\) , \( a - 3\) , \( -a + 1\bigr] \)
4.1-a5 \( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 3\) , \( -a + 4\bigr] \)
4.1-a6 \( \bigl[1\) , \( 1\) , \( a + 1\) , \( 28 a - 27\) , \( 51 a - 158\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 45 & 3 & 5 & 15 & 9 \\ 45 & 1 & 15 & 9 & 3 & 5 \\ 3 & 15 & 1 & 15 & 5 & 3 \\ 5 & 9 & 15 & 1 & 3 & 45 \\ 15 & 3 & 5 & 3 & 1 & 15 \\ 9 & 5 & 3 & 45 & 15 & 1 \end{array}\right)\)

Isogeny graph