Properties

Base field \(\Q(\zeta_{13})^+\)
Label 6.6.371293.1-79.3-c
Conductor 79.3
Rank \( 0 \)

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Base field \(\Q(\zeta_{13})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).

Elliptic curves in class 79.3-c over \(\Q(\zeta_{13})^+\)

Isogeny class 79.3-c contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
79.3-c1 \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{4} - 3 a^{2} - a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( -57 a^{5} + 16 a^{4} + 293 a^{3} - 17 a^{2} - 350 a - 80\) , \( 390 a^{5} - 113 a^{4} - 2030 a^{3} + 120 a^{2} + 2423 a + 548\bigr] \)
79.3-c2 \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( -a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a - 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( 15 a^{5} + 9 a^{4} - 64 a^{3} - 43 a^{2} + 37 a + 19\) , \( -56 a^{5} - 29 a^{4} + 239 a^{3} + 138 a^{2} - 138 a - 42\bigr] \)
79.3-c3 \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 4\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 3\) , \( 11 a^{5} - 45 a^{4} - 15 a^{3} + 102 a^{2} + 9 a - 54\) , \( 59 a^{5} - 254 a^{4} - 57 a^{3} + 539 a^{2} + 2 a - 247\bigr] \)
79.3-c4 \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{5} + 5 a^{3} - a^{2} - 6 a + 2\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a\) , \( -435 a^{5} - 121 a^{4} + 1512 a^{3} + 155 a^{2} - 866 a - 246\) , \( -8307 a^{5} - 5881 a^{4} + 27378 a^{3} + 12730 a^{2} - 16010 a - 4422\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 10 & 5 \\ 2 & 1 & 5 & 10 \\ 10 & 5 & 1 & 2 \\ 5 & 10 & 2 & 1 \end{array}\right)\)

Isogeny graph