Properties

Base field 4.4.6809.1
Label 4.4.6809.1-10.1-a
Conductor 10.1
Rank \( 0 \)

Related objects

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Base field 4.4.6809.1

Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} - x + 1 \); class number \(1\).

Elliptic curves in class 10.1-a over 4.4.6809.1

Isogeny class 10.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
10.1-a1 \( \bigl[a^{3} + a^{2} - 5 a - 4\) , \( -a^{2} - a + 4\) , \( a^{3} + a^{2} - 4 a - 3\) , \( -8 a^{3} - 63 a^{2} - 56 a - 7\) , \( -7157 a^{3} - 16545 a^{2} - 2241 a + 2899\bigr] \)
10.1-a2 \( \bigl[a^{2} + a - 3\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 4 a\) , \( 52 a^{3} + 8 a^{2} - 277 a - 149\) , \( 268 a^{3} + 49 a^{2} - 1434 a - 803\bigr] \)
10.1-a3 \( \bigl[a^{2} + a - 3\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 4 a\) , \( 12 a^{3} + 3 a^{2} - 57 a - 29\) , \( 23 a^{3} + 13 a^{2} - 103 a - 61\bigr] \)
10.1-a4 \( \bigl[a^{2} + a - 3\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 4 a\) , \( 2 a^{3} - 2 a^{2} - 12 a + 1\) , \( 0\bigr] \)
10.1-a5 \( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + 2 a - 2\) , \( 2 a^{3} - 7 a + 2\bigr] \)
10.1-a6 \( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a^{2} + a - 2\) , \( 10 a^{3} - 19 a^{2} - 8 a + 3\) , \( a^{3} - 72 a^{2} + 186 a - 59\bigr] \)
10.1-a7 \( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 6 a - 1\) , \( a^{2} + a - 3\) , \( -a^{3} + 3 a^{2} + 4 a - 11\) , \( -10 a^{3} + 4 a^{2} + 46 a - 10\bigr] \)
10.1-a8 \( \bigl[1\) , \( -a^{3} + 5 a\) , \( 1\) , \( -37 a^{3} + 16 a^{2} + 161 a - 59\) , \( -116 a^{3} + 55 a^{2} + 605 a - 222\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph