Elliptic curve isogeny class 27.1-a over number field \(\Q(\sqrt{13}) \)

• Base field: \(\Q(\sqrt{13}) \)
• Label: 2.2.13.1-27.1-a
• Conductor: 27.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 27.1-a over \(\Q(\sqrt{13}) \)

Isogeny class 27.1-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label	Weierstrass Coefficients
27.1-a1	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -a + 1\) , \( -2\bigr] \)
27.1-a2	\( \bigl[a\) , \( 0\) , \( a\) , \( -59 a - 135\) , \( 766 a + 829\bigr] \)
27.1-a3	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 464 a - 989\) , \( 6420 a - 15050\bigr] \)
27.1-a4	\( \bigl[a\) , \( 0\) , \( a\) , \( a\) , \( a + 1\bigr] \)
27.1-a5	\( \bigl[a\) , \( 0\) , \( a\) , \( -4 a - 15\) , \( a + 10\bigr] \)
27.1-a6	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 19 a - 74\) , \( 70 a - 224\bigr] \)
27.1-a7	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -26 a - 119\) , \( 340 a + 10\bigr] \)
27.1-a8	\( \bigl[a\) , \( 0\) , \( a\) , \( -4 a - 150\) , \( -404 a + 118\bigr] \)
27.1-a9	\( \bigl[a\) , \( 0\) , \( a\) , \( -84 a - 120\) , \( 546 a + 718\bigr] \)
27.1-a10	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -6 a - 14\) , \( -20 a - 26\bigr] \)
27.1-a11	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -847 a + 1914\) , \( -19366 a + 44646\bigr] \)
27.1-a12	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -111 a - 194\) , \( -1130 a - 1304\bigr] \)

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