*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
Label |
Base field |
Conductor |
Isogeny class |
Weierstrass coefficients |
27.1-a1 |
\(\Q(\zeta_{7})^+\)
|
27.1 |
27.1-a |
\( \bigl[0\) , \( -a\) , \( a\) , \( 652 a^{2} - 391 a - 1564\) , \( 10528 a^{2} - 5979 a - 24046\bigr] \) |
27.1-a2 |
\(\Q(\zeta_{7})^+\)
|
27.1 |
27.1-a |
\( \bigl[0\) , \( -a\) , \( a\) , \( 2 a^{2} - a - 4\) , \( -2 a^{2} + a + 4\bigr] \) |
41.1-a1 |
\(\Q(\zeta_{7})^+\)
|
41.1 |
41.1-a |
\( \bigl[1\) , \( -a^{2} + 3\) , \( 1\) , \( 99 a^{2} - 10 a - 348\) , \( 952 a^{2} - 216 a - 2798\bigr] \) |
41.1-a2 |
\(\Q(\zeta_{7})^+\)
|
41.1 |
41.1-a |
\( \bigl[1\) , \( -a^{2} + 3\) , \( 1\) , \( 144 a^{2} - 75 a - 328\) , \( 1172 a^{2} - 646 a - 2650\bigr] \) |
41.1-a3 |
\(\Q(\zeta_{7})^+\)
|
41.1 |
41.1-a |
\( \bigl[1\) , \( -a^{2} + 3\) , \( 1\) , \( -a^{2} + 2\) , \( 0\bigr] \) |
41.1-a4 |
\(\Q(\zeta_{7})^+\)
|
41.1 |
41.1-a |
\( \bigl[1\) , \( -a^{2} + 3\) , \( 1\) , \( -6 a^{2} + 7\) , \( 2 a^{2} + 4 a + 2\bigr] \) |
41.2-a1 |
\(\Q(\zeta_{7})^+\)
|
41.2 |
41.2-a |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -10 a^{2} - 89 a - 140\) , \( -216 a^{2} - 736 a - 678\bigr] \) |
41.2-a2 |
\(\Q(\zeta_{7})^+\)
|
41.2 |
41.2-a |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 6 a - 5\) , \( 4 a^{2} - 6 a + 2\bigr] \) |
41.2-a3 |
\(\Q(\zeta_{7})^+\)
|
41.2 |
41.2-a |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( a\) , \( 0\bigr] \) |
41.2-a4 |
\(\Q(\zeta_{7})^+\)
|
41.2 |
41.2-a |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -75 a^{2} - 69 a + 35\) , \( -646 a^{2} - 526 a + 340\bigr] \) |
41.3-a1 |
\(\Q(\zeta_{7})^+\)
|
41.3 |
41.3-a |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( -91 a^{2} + 101 a - 68\) , \( -646 a^{2} + 852 a - 304\bigr] \) |
41.3-a2 |
\(\Q(\zeta_{7})^+\)
|
41.3 |
41.3-a |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( -71 a^{2} + 146 a - 43\) , \( -456 a^{2} + 1027 a - 381\bigr] \) |
41.3-a3 |
\(\Q(\zeta_{7})^+\)
|
41.3 |
41.3-a |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( 4 a^{2} - 4 a - 8\) , \( -11 a^{2} + 7 a + 26\bigr] \) |
41.3-a4 |
\(\Q(\zeta_{7})^+\)
|
41.3 |
41.3-a |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( -a^{2} + a + 2\) , \( 0\bigr] \) |
49.1-a1 |
\(\Q(\zeta_{7})^+\)
|
49.1 |
49.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 3\) , \( a + 1\) , \( 62 a^{2} - 26 a - 156\) , \( -380 a^{2} + 192 a + 886\bigr] \) |
49.1-a2 |
\(\Q(\zeta_{7})^+\)
|
49.1 |
49.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 3\) , \( a + 1\) , \( 2 a^{2} - a - 6\) , \( -9 a^{2} + 4 a + 20\bigr] \) |
49.1-a3 |
\(\Q(\zeta_{7})^+\)
|
49.1 |
49.1-a |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -37\) , \( -78\bigr] \) |
49.1-a4 |
\(\Q(\zeta_{7})^+\)
|
49.1 |
49.1-a |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -2\) , \( -1\bigr] \) |
56.1-a1 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
56.1-a2 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
56.1-a3 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
56.1-a4 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
56.1-a5 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
56.1-a6 |
\(\Q(\zeta_{7})^+\)
|
56.1 |
56.1-a |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
64.1-a1 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1742 a^{2} - 971 a - 3929\) , \( 42399 a^{2} - 23535 a - 95292\bigr] \) |
64.1-a2 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 72 a^{2} + 4 a - 264\) , \( 500 a^{2} - 16 a - 1624\bigr] \) |
64.1-a3 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 102 a^{2} - 61 a - 244\) , \( 640 a^{2} - 360 a - 1460\bigr] \) |
64.1-a4 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 17 a^{2} - 131 a - 199\) , \( -6 a^{2} - 855 a - 1073\bigr] \) |
64.1-a5 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 22 a^{2} - 11 a - 49\) , \( 55 a^{2} - 31 a - 124\bigr] \) |
64.1-a6 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 17 a^{2} - 11 a - 39\) , \( -46 a^{2} + 25 a + 103\bigr] \) |
64.1-a7 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 2 a^{2} - a - 4\) , \( 0\bigr] \) |
64.1-a8 |
\(\Q(\zeta_{7})^+\)
|
64.1 |
64.1-a |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -8 a^{2} + 4 a + 16\) , \( 4 a^{2} - 8\bigr] \) |
71.1-a1 |
\(\Q(\zeta_{7})^+\)
|
71.1 |
71.1-a |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -190 a^{2} + 410 a - 171\) , \( -2384 a^{2} + 5309 a - 1936\bigr] \) |
71.1-a2 |
\(\Q(\zeta_{7})^+\)
|
71.1 |
71.1-a |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -10 a^{2} + 5 a - 26\) , \( -61 a^{2} + 92 a - 76\bigr] \) |
71.1-a3 |
\(\Q(\zeta_{7})^+\)
|
71.1 |
71.1-a |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -5 a^{2} + 15 a - 11\) , \( 12 a^{2} - 31 a + 15\bigr] \) |
71.1-a4 |
\(\Q(\zeta_{7})^+\)
|
71.1 |
71.1-a |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -1\) , \( -a\bigr] \) |
71.2-a1 |
\(\Q(\zeta_{7})^+\)
|
71.2 |
71.2-a |
\( \bigl[1\) , \( -a^{2} + a + 1\) , \( a^{2} + a - 2\) , \( 410 a^{2} - 221 a - 960\) , \( 5309 a^{2} - 2926 a - 12013\bigr] \) |
71.2-a2 |
\(\Q(\zeta_{7})^+\)
|
71.2 |
71.2-a |
\( \bigl[1\) , \( -a^{2} + a + 1\) , \( a^{2} + a - 2\) , \( 5 a^{2} + 4 a - 50\) , \( 92 a^{2} - 32 a - 290\bigr] \) |
71.2-a3 |
\(\Q(\zeta_{7})^+\)
|
71.2 |
71.2-a |
\( \bigl[1\) , \( -a^{2} + a + 1\) , \( a^{2} + a - 2\) , \( 15 a^{2} - 11 a - 35\) , \( -31 a^{2} + 18 a + 70\bigr] \) |
71.2-a4 |
\(\Q(\zeta_{7})^+\)
|
71.2 |
71.2-a |
\( \bigl[1\) , \( -a^{2} + a + 1\) , \( a^{2} + a - 2\) , \( -a\) , \( -a^{2} + 1\bigr] \) |
71.3-a1 |
\(\Q(\zeta_{7})^+\)
|
71.3 |
71.3-a |
\( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} - 1\) , \( 4 a^{2} - 10 a - 44\) , \( -32 a^{2} - 61 a - 73\bigr] \) |
71.3-a2 |
\(\Q(\zeta_{7})^+\)
|
71.3 |
71.3-a |
\( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} - 1\) , \( -a^{2} + 1\) , \( 0\bigr] \) |
71.3-a3 |
\(\Q(\zeta_{7})^+\)
|
71.3 |
71.3-a |
\( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} - 1\) , \( -11 a^{2} - 5 a + 6\) , \( 18 a^{2} + 12 a - 9\bigr] \) |
71.3-a4 |
\(\Q(\zeta_{7})^+\)
|
71.3 |
71.3-a |
\( \bigl[1\) , \( a^{2} - 2\) , \( a^{2} - 1\) , \( -221 a^{2} - 190 a + 81\) , \( -2926 a^{2} - 2384 a + 1532\bigr] \) |
91.1-a1 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -1030 a^{2} + 1620 a - 484\) , \( -21769 a^{2} + 41147 a - 14213\bigr] \) |
91.1-a2 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( 15 a^{2} + 5 a - 89\) , \( -89 a^{2} - 20 a + 385\bigr] \) |
91.1-a3 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -4\) , \( -3 a^{2} - a + 9\bigr] \) |
91.1-a4 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -15 a^{2} - 5 a + 1\) , \( -49 a^{2} - 26 a + 29\bigr] \) |
91.1-a5 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
91.1-a6 |
\(\Q(\zeta_{7})^+\)
|
91.1 |
91.1-a |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -10 a^{2} - 20 a + 21\) , \( -90 a^{2} + 20 a + 26\bigr] \) |