Elliptic curve isogeny class 405.1-c over number field \(\Q(\zeta_{20})^+\)

• Base field: \(\Q(\zeta_{20})^+\)
• Label: 4.4.2000.1-405.1-c
• Conductor: 405.1
• Rank: \( 0 \)

Base field \(\Q(\zeta_{20})^+\)

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

Elliptic curves in class 405.1-c over \(\Q(\zeta_{20})^+\)

Isogeny class 405.1-c contains 10 curves linked by isogenies of degrees dividing 32.

Curve label	Weierstrass Coefficients
405.1-c1	\( \bigl[a^{2} - 3\) , \( -a^{2} + 4\) , \( 1\) , \( 4363 a^{2} - 20828\) , \( 255406 a^{2} - 1062683\bigr] \)
405.1-c2	\( \bigl[a^{2} - 2\) , \( a^{2} - 4\) , \( a^{2} - 3\) , \( -4366 a^{2} + 993\) , \( -251041 a^{2} + 213356\bigr] \)
405.1-c3	\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \)
405.1-c4	\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \)
405.1-c5	\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \)
405.1-c6	\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \)
405.1-c7	\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \)
405.1-c8	\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \)
405.1-c9	\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \)
405.1-c10	\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrr} 1 & 4 & 2 & 4 & 8 & 32 & 8 & 16 & 32 & 16 \\ 4 & 1 & 2 & 4 & 8 & 32 & 8 & 16 & 32 & 16 \\ 2 & 2 & 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 4 & 4 & 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 8 & 8 & 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 32 & 32 & 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 8 & 8 & 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 16 & 16 & 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 32 & 32 & 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 16 & 16 & 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph