Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-841.8-a
Conductor 841.8
Rank \( 0 \)

Related objects

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Base field \(\Q(\zeta_{15})^+\)

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

Elliptic curves in class 841.8-a over \(\Q(\zeta_{15})^+\)

Isogeny class 841.8-a contains 8 curves linked by isogenies of degrees dividing 30.

Curve label Weierstrass Coefficients
841.8-a1 \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 732862 a^{3} + 606139 a^{2} - 1823977 a - 401107\) , \( 617507703 a^{3} + 510735017 a^{2} - 1536871501 a - 337973166\bigr] \)
841.8-a2 \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -4944643 a^{3} - 4089726 a^{2} + 12306253 a + 2706273\) , \( 6260338923 a^{3} + 5177869806 a^{2} - 15580916689 a - 3426396906\bigr] \)
841.8-a3 \( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 46 a^{3} + 3 a^{2} - 179 a - 37\) , \( 315 a^{3} + 91 a^{2} - 1189 a - 251\bigr] \)
841.8-a4 \( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 6 a^{3} + 3 a^{2} - 19 a - 2\) , \( 3 a^{3} - a^{2} - 16 a - 4\bigr] \)
841.8-a5 \( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -378019 a^{3} - 312648 a^{2} + 940838 a + 206900\) , \( -174410939 a^{3} - 144253725 a^{2} + 434079076 a + 95458261\bigr] \)
841.8-a6 \( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -6055524 a^{3} - 5008513 a^{2} + 15071068 a + 3314280\) , \( -11131060964 a^{3} - 9206399587 a^{2} + 27703314743 a + 6092231498\bigr] \)
841.8-a7 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -70 a^{3} - 57 a^{2} + 174 a + 37\) , \( 432 a^{3} + 358 a^{2} - 1075 a - 238\bigr] \)
841.8-a8 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -1125 a^{3} - 897 a^{2} + 2819 a + 527\) , \( 27556 a^{3} + 22887 a^{2} - 68523 a - 15332\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 30 & 15 & 5 & 10 & 3 & 6 \\ 2 & 1 & 15 & 30 & 10 & 5 & 6 & 3 \\ 30 & 15 & 1 & 2 & 6 & 3 & 10 & 5 \\ 15 & 30 & 2 & 1 & 3 & 6 & 5 & 10 \\ 5 & 10 & 6 & 3 & 1 & 2 & 15 & 30 \\ 10 & 5 & 3 & 6 & 2 & 1 & 30 & 15 \\ 3 & 6 & 10 & 5 & 15 & 30 & 1 & 2 \\ 6 & 3 & 5 & 10 & 30 & 15 & 2 & 1 \end{array}\right)\)

Isogeny graph