Properties

Base field 4.4.10512.1
Label 4.4.10512.1-4.1-b
Conductor 4.1
Rank not recorded

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Base field 4.4.10512.1

Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).

Elliptic curves in class 4.1-b over 4.4.10512.1

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

Curve label Weierstrass Coefficients
4.1-b1 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 589 a^{3} + 153 a^{2} - 4324 a - 4830\) , \( -18547 a^{3} - 3626 a^{2} + 131800 a + 137439\bigr] \)
4.1-b2 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -96 a^{3} + 123 a^{2} + 526 a - 90\) , \( 723 a^{3} - 903 a^{2} - 3931 a + 590\bigr] \)
4.1-b3 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 29 a^{3} - 27 a^{2} - 144 a - 50\) , \( -11 a^{3} - 34 a^{2} + 184 a + 207\bigr] \)
4.1-b4 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} - 22 a^{2} - 39 a - 10\) , \( -55 a^{3} - 195 a^{2} - 161 a + 14\bigr] \)
4.1-b5 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} + 3 a^{2} + 6 a - 5\) , \( a^{3} - a^{2} - 5 a\bigr] \)
4.1-b6 \( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 6 a - 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 689 a^{3} - 1427 a^{2} - 2064 a + 250\) , \( 22605 a^{3} - 44138 a^{2} - 74800 a + 11751\bigr] \)

Rank

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Isogeny matrix

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

Isogeny graph