Properties

Base field 4.4.11324.1
Label 4.4.11324.1-10.1-b
Conductor 10.1
Rank \( 1 \)

Related objects

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Base field 4.4.11324.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 4 x + 2 \); class number \(1\).

Elliptic curves in class 10.1-b over 4.4.11324.1

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

Curve label Weierstrass Coefficients
10.1-b1 \( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} - 4 a\) , \( -29 a^{3} - 35 a^{2} + 67 a + 28\) , \( -250 a^{3} - 312 a^{2} + 546 a + 222\bigr] \)
10.1-b2 \( \bigl[a^{2} - 3\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{2} + a - 3\) , \( 172 a^{3} - 551 a^{2} + 326 a + 110\) , \( 4341 a^{3} - 13428 a^{2} + 6464 a + 3927\bigr] \)
10.1-b3 \( \bigl[a^{2} - 3\) , \( a^{2} - 2\) , \( a\) , \( 24 a^{3} + 15 a^{2} - 118 a - 98\) , \( 175 a^{3} + 12 a^{2} - 839 a - 228\bigr] \)
10.1-b4 \( \bigl[a^{2} - 3\) , \( a^{2} - 2\) , \( a\) , \( -6 a^{3} + 10 a^{2} + 27 a - 38\) , \( 17 a^{3} - 22 a^{2} - 76 a + 92\bigr] \)
10.1-b5 \( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( -6 a^{3} - 2 a^{2} + 5 a + 5\) , \( 12 a^{3} + 9 a^{2} - 20 a - 9\bigr] \)
10.1-b6 \( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( -a^{3} - 2 a^{2} + 5 a + 5\) , \( -a^{3} + a^{2} + 2 a - 1\bigr] \)
10.1-b7 \( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( 4 a^{3} - 82 a^{2} + 60 a + 35\) , \( 73 a^{3} - 606 a^{2} + 424 a + 227\bigr] \)
10.1-b8 \( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( 4 a^{3} - 82 a^{2} + 65 a + 35\) , \( 56 a^{3} - 593 a^{2} + 430 a + 227\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph