Properties

Base field 4.4.4400.1
Label 4.4.4400.1-31.2-c
Conductor 31.2
Rank \( 1 \)

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Base field 4.4.4400.1

Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} + 11 \); class number \(1\).

Elliptic curves in class 31.2-c over 4.4.4400.1

Isogeny class 31.2-c contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
31.2-c1 \( \bigl[a^{3} + a^{2} - 4 a - 4\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} - 3 a\) , \( -116 a^{3} + 88 a^{2} + 840 a - 1067\) , \( 2720 a^{3} - 3047 a^{2} - 16831 a + 23245\bigr] \)
31.2-c2 \( \bigl[a^{2} + a - 4\) , \( -a^{2} + 4\) , \( a^{3} - 4 a\) , \( a^{3} - 3 a^{2} - 5 a + 12\) , \( 30 a^{3} + 64 a^{2} - 72 a - 153\bigr] \)
31.2-c3 \( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} + a^{2} - 4 a - 4\) , \( 80 a^{3} - 151 a^{2} - 203 a + 339\) , \( -712 a^{3} + 1513 a^{2} + 1703 a - 3582\bigr] \)
31.2-c4 \( \bigl[1\) , \( a^{2} - a - 3\) , \( a^{3} + a^{2} - 4 a - 4\) , \( 1477 a^{3} + 2281 a^{2} - 6825 a - 10537\) , \( 49010 a^{3} + 75642 a^{2} - 226327 a - 349309\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 4 & 4 & 2 \\ 4 & 1 & 4 & 2 \\ 4 & 4 & 1 & 2 \\ 2 & 2 & 2 & 1 \end{array}\right)\)

Isogeny graph