Normalized defining polynomial
\( x^{21} - 2 x^{20} - 32 x^{19} + 51 x^{18} + 432 x^{17} - 473 x^{16} - 3214 x^{15} + 1767 x^{14} + \cdots + 8 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[21, 0]$ |
| |
| Discriminant: |
\(2406787169604002863343075235725312\)
\(\medspace = 2^{27}\cdot 7^{14}\cdot 31^{9}\)
|
| |
| Root discriminant: | \(38.87\) |
| |
| Galois root discriminant: | $2^{3/2}7^{2/3}31^{1/2}\approx 57.626803948274514$ | ||
| Ramified primes: |
\(2\), \(7\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{62}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{10\cdots 12}a^{20}-\frac{7949774923281}{10\cdots 12}a^{19}+\frac{12715609328879}{269894553023353}a^{18}+\frac{184189865438001}{10\cdots 12}a^{17}-\frac{84175519522045}{10\cdots 12}a^{16}-\frac{21304611023181}{10\cdots 12}a^{15}-\frac{227835955447891}{10\cdots 12}a^{14}-\frac{5738938604513}{25106470048684}a^{13}+\frac{449140295544647}{10\cdots 12}a^{12}-\frac{40692109083229}{10\cdots 12}a^{11}+\frac{125643006466249}{10\cdots 12}a^{10}+\frac{466747363165087}{10\cdots 12}a^{9}+\frac{143955032155503}{10\cdots 12}a^{8}-\frac{395339627893689}{10\cdots 12}a^{7}+\frac{6779693814587}{539789106046706}a^{6}+\frac{85957087125313}{539789106046706}a^{5}+\frac{506684194022259}{10\cdots 12}a^{4}-\frac{8811130171145}{539789106046706}a^{3}+\frac{120217707214026}{269894553023353}a^{2}-\frac{63940930981172}{269894553023353}a+\frac{80943118923147}{269894553023353}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $20$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{7886249775825}{302487590948}a^{20}-\frac{21396779684043}{302487590948}a^{19}-\frac{118578700243223}{151243795474}a^{18}+\frac{571577089820901}{302487590948}a^{17}+\frac{30\cdots 85}{302487590948}a^{16}-\frac{58\cdots 39}{302487590948}a^{15}-\frac{21\cdots 59}{302487590948}a^{14}+\frac{29\cdots 55}{302487590948}a^{13}+\frac{90\cdots 99}{302487590948}a^{12}-\frac{67\cdots 39}{302487590948}a^{11}-\frac{23\cdots 95}{302487590948}a^{10}+\frac{41\cdots 21}{302487590948}a^{9}+\frac{31\cdots 63}{302487590948}a^{8}+\frac{73\cdots 41}{302487590948}a^{7}-\frac{83\cdots 77}{151243795474}a^{6}-\frac{35\cdots 15}{151243795474}a^{5}+\frac{29\cdots 93}{302487590948}a^{4}+\frac{35\cdots 24}{75621897737}a^{3}-\frac{989343207867421}{151243795474}a^{2}-\frac{182143529681500}{75621897737}a+\frac{21517905640571}{75621897737}$, $\frac{7886249775825}{302487590948}a^{20}-\frac{21396779684043}{302487590948}a^{19}-\frac{118578700243223}{151243795474}a^{18}+\frac{571577089820901}{302487590948}a^{17}+\frac{30\cdots 85}{302487590948}a^{16}-\frac{58\cdots 39}{302487590948}a^{15}-\frac{21\cdots 59}{302487590948}a^{14}+\frac{29\cdots 55}{302487590948}a^{13}+\frac{90\cdots 99}{302487590948}a^{12}-\frac{67\cdots 39}{302487590948}a^{11}-\frac{23\cdots 95}{302487590948}a^{10}+\frac{41\cdots 21}{302487590948}a^{9}+\frac{31\cdots 63}{302487590948}a^{8}+\frac{73\cdots 41}{302487590948}a^{7}-\frac{83\cdots 77}{151243795474}a^{6}-\frac{35\cdots 15}{151243795474}a^{5}+\frac{29\cdots 93}{302487590948}a^{4}+\frac{35\cdots 24}{75621897737}a^{3}-\frac{989343207867421}{151243795474}a^{2}-\frac{182143529681500}{75621897737}a+\frac{21593527538308}{75621897737}$, $\frac{21\cdots 37}{10\cdots 12}a^{20}-\frac{14\cdots 01}{269894553023353}a^{19}-\frac{31\cdots 25}{539789106046706}a^{18}+\frac{15\cdots 79}{10\cdots 12}a^{17}+\frac{40\cdots 91}{539789106046706}a^{16}-\frac{15\cdots 61}{10\cdots 12}a^{15}-\frac{28\cdots 85}{539789106046706}a^{14}+\frac{18\cdots 37}{25106470048684}a^{13}+\frac{60\cdots 04}{269894553023353}a^{12}-\frac{18\cdots 37}{10\cdots 12}a^{11}-\frac{30\cdots 95}{539789106046706}a^{10}+\frac{11\cdots 05}{10\cdots 12}a^{9}+\frac{41\cdots 99}{539789106046706}a^{8}+\frac{19\cdots 01}{10\cdots 12}a^{7}-\frac{44\cdots 71}{10\cdots 12}a^{6}-\frac{18\cdots 97}{10\cdots 12}a^{5}+\frac{39\cdots 47}{539789106046706}a^{4}+\frac{94\cdots 87}{269894553023353}a^{3}-\frac{26\cdots 33}{539789106046706}a^{2}-\frac{48\cdots 11}{269894553023353}a+\frac{58\cdots 34}{269894553023353}$, $\frac{9342976803487}{151243795474}a^{20}-\frac{12764982886622}{75621897737}a^{19}-\frac{280241039373715}{151243795474}a^{18}+\frac{681655210710877}{151243795474}a^{17}+\frac{35\cdots 11}{151243795474}a^{16}-\frac{70\cdots 33}{151243795474}a^{15}-\frac{12\cdots 59}{75621897737}a^{14}+\frac{34\cdots 35}{151243795474}a^{13}+\frac{53\cdots 62}{75621897737}a^{12}-\frac{80\cdots 23}{151243795474}a^{11}-\frac{13\cdots 73}{75621897737}a^{10}+\frac{51\cdots 31}{151243795474}a^{9}+\frac{18\cdots 79}{75621897737}a^{8}+\frac{84\cdots 25}{151243795474}a^{7}-\frac{19\cdots 25}{151243795474}a^{6}-\frac{83\cdots 17}{151243795474}a^{5}+\frac{34\cdots 61}{151243795474}a^{4}+\frac{83\cdots 85}{75621897737}a^{3}-\frac{23\cdots 67}{151243795474}a^{2}-\frac{428524642333825}{75621897737}a+\frac{50944432924518}{75621897737}$, $\frac{15\cdots 49}{10\cdots 12}a^{20}-\frac{41\cdots 43}{10\cdots 12}a^{19}-\frac{11\cdots 97}{269894553023353}a^{18}+\frac{11\cdots 27}{10\cdots 12}a^{17}+\frac{58\cdots 19}{10\cdots 12}a^{16}-\frac{11\cdots 91}{10\cdots 12}a^{15}-\frac{41\cdots 05}{10\cdots 12}a^{14}+\frac{13\cdots 57}{25106470048684}a^{13}+\frac{17\cdots 41}{10\cdots 12}a^{12}-\frac{13\cdots 35}{10\cdots 12}a^{11}-\frac{45\cdots 13}{10\cdots 12}a^{10}+\frac{84\cdots 01}{10\cdots 12}a^{9}+\frac{61\cdots 97}{10\cdots 12}a^{8}+\frac{13\cdots 73}{10\cdots 12}a^{7}-\frac{82\cdots 22}{269894553023353}a^{6}-\frac{68\cdots 41}{539789106046706}a^{5}+\frac{58\cdots 37}{10\cdots 12}a^{4}+\frac{13\cdots 03}{539789106046706}a^{3}-\frac{19\cdots 77}{539789106046706}a^{2}-\frac{35\cdots 60}{269894553023353}a+\frac{42\cdots 95}{269894553023353}$, $\frac{10\cdots 47}{539789106046706}a^{20}-\frac{56\cdots 81}{10\cdots 12}a^{19}-\frac{30\cdots 07}{539789106046706}a^{18}+\frac{37\cdots 41}{269894553023353}a^{17}+\frac{77\cdots 11}{10\cdots 12}a^{16}-\frac{38\cdots 57}{269894553023353}a^{15}-\frac{54\cdots 81}{10\cdots 12}a^{14}+\frac{88\cdots 29}{12553235024342}a^{13}+\frac{23\cdots 47}{10\cdots 12}a^{12}-\frac{44\cdots 96}{269894553023353}a^{11}-\frac{59\cdots 01}{10\cdots 12}a^{10}+\frac{55\cdots 53}{539789106046706}a^{9}+\frac{80\cdots 81}{10\cdots 12}a^{8}+\frac{93\cdots 71}{539789106046706}a^{7}-\frac{43\cdots 53}{10\cdots 12}a^{6}-\frac{18\cdots 53}{10\cdots 12}a^{5}+\frac{75\cdots 05}{10\cdots 12}a^{4}+\frac{92\cdots 90}{269894553023353}a^{3}-\frac{12\cdots 92}{269894553023353}a^{2}-\frac{47\cdots 72}{269894553023353}a+\frac{56\cdots 54}{269894553023353}$, $\frac{18\cdots 43}{539789106046706}a^{20}-\frac{98\cdots 19}{10\cdots 12}a^{19}-\frac{55\cdots 49}{539789106046706}a^{18}+\frac{65\cdots 10}{269894553023353}a^{17}+\frac{14\cdots 77}{10\cdots 12}a^{16}-\frac{67\cdots 07}{269894553023353}a^{15}-\frac{10\cdots 93}{10\cdots 12}a^{14}+\frac{15\cdots 29}{12553235024342}a^{13}+\frac{43\cdots 99}{10\cdots 12}a^{12}-\frac{76\cdots 42}{269894553023353}a^{11}-\frac{11\cdots 01}{10\cdots 12}a^{10}+\frac{87\cdots 69}{539789106046706}a^{9}+\frac{15\cdots 29}{10\cdots 12}a^{8}+\frac{18\cdots 05}{539789106046706}a^{7}-\frac{80\cdots 17}{10\cdots 12}a^{6}-\frac{35\cdots 71}{10\cdots 12}a^{5}+\frac{14\cdots 05}{10\cdots 12}a^{4}+\frac{17\cdots 21}{269894553023353}a^{3}-\frac{48\cdots 79}{539789106046706}a^{2}-\frac{89\cdots 08}{269894553023353}a+\frac{10\cdots 71}{269894553023353}$, $\frac{23\cdots 55}{269894553023353}a^{20}-\frac{24\cdots 63}{10\cdots 12}a^{19}-\frac{69\cdots 52}{269894553023353}a^{18}+\frac{33\cdots 81}{539789106046706}a^{17}+\frac{35\cdots 49}{10\cdots 12}a^{16}-\frac{34\cdots 53}{539789106046706}a^{15}-\frac{24\cdots 29}{10\cdots 12}a^{14}+\frac{19\cdots 32}{6276617512171}a^{13}+\frac{10\cdots 35}{10\cdots 12}a^{12}-\frac{39\cdots 43}{539789106046706}a^{11}-\frac{27\cdots 81}{10\cdots 12}a^{10}+\frac{11\cdots 88}{269894553023353}a^{9}+\frac{37\cdots 49}{10\cdots 12}a^{8}+\frac{22\cdots 77}{269894553023353}a^{7}-\frac{19\cdots 19}{10\cdots 12}a^{6}-\frac{84\cdots 69}{10\cdots 12}a^{5}+\frac{34\cdots 39}{10\cdots 12}a^{4}+\frac{42\cdots 42}{269894553023353}a^{3}-\frac{11\cdots 67}{539789106046706}a^{2}-\frac{21\cdots 78}{269894553023353}a+\frac{26\cdots 41}{269894553023353}$, $\frac{51\cdots 83}{10\cdots 12}a^{20}-\frac{14\cdots 89}{10\cdots 12}a^{19}-\frac{77\cdots 47}{539789106046706}a^{18}+\frac{37\cdots 49}{10\cdots 12}a^{17}+\frac{19\cdots 17}{10\cdots 12}a^{16}-\frac{38\cdots 97}{10\cdots 12}a^{15}-\frac{13\cdots 85}{10\cdots 12}a^{14}+\frac{44\cdots 23}{25106470048684}a^{13}+\frac{59\cdots 81}{10\cdots 12}a^{12}-\frac{44\cdots 65}{10\cdots 12}a^{11}-\frac{15\cdots 21}{10\cdots 12}a^{10}+\frac{28\cdots 67}{10\cdots 12}a^{9}+\frac{20\cdots 61}{10\cdots 12}a^{8}+\frac{47\cdots 31}{10\cdots 12}a^{7}-\frac{54\cdots 63}{539789106046706}a^{6}-\frac{23\cdots 51}{539789106046706}a^{5}+\frac{19\cdots 63}{10\cdots 12}a^{4}+\frac{46\cdots 77}{539789106046706}a^{3}-\frac{32\cdots 86}{269894553023353}a^{2}-\frac{11\cdots 47}{269894553023353}a+\frac{14\cdots 57}{269894553023353}$, $\frac{33577707494391}{151243795474}a^{20}-\frac{45695072839843}{75621897737}a^{19}-\frac{504313229957035}{75621897737}a^{18}+\frac{12\cdots 48}{75621897737}a^{17}+\frac{12\cdots 93}{151243795474}a^{16}-\frac{25\cdots 25}{151243795474}a^{15}-\frac{44\cdots 41}{75621897737}a^{14}+\frac{12\cdots 61}{151243795474}a^{13}+\frac{19\cdots 51}{75621897737}a^{12}-\frac{28\cdots 33}{151243795474}a^{11}-\frac{49\cdots 26}{75621897737}a^{10}+\frac{17\cdots 73}{151243795474}a^{9}+\frac{66\cdots 04}{75621897737}a^{8}+\frac{30\cdots 73}{151243795474}a^{7}-\frac{71\cdots 69}{151243795474}a^{6}-\frac{30\cdots 57}{151243795474}a^{5}+\frac{62\cdots 16}{75621897737}a^{4}+\frac{60\cdots 01}{151243795474}a^{3}-\frac{85\cdots 99}{151243795474}a^{2}-\frac{15\cdots 92}{75621897737}a+\frac{185255262902082}{75621897737}$, $\frac{84\cdots 29}{10\cdots 12}a^{20}-\frac{57\cdots 19}{269894553023353}a^{19}-\frac{12\cdots 89}{539789106046706}a^{18}+\frac{61\cdots 45}{10\cdots 12}a^{17}+\frac{15\cdots 61}{539789106046706}a^{16}-\frac{63\cdots 81}{10\cdots 12}a^{15}-\frac{56\cdots 58}{269894553023353}a^{14}+\frac{73\cdots 13}{25106470048684}a^{13}+\frac{48\cdots 89}{539789106046706}a^{12}-\frac{72\cdots 01}{10\cdots 12}a^{11}-\frac{61\cdots 94}{269894553023353}a^{10}+\frac{46\cdots 29}{10\cdots 12}a^{9}+\frac{82\cdots 25}{269894553023353}a^{8}+\frac{75\cdots 73}{10\cdots 12}a^{7}-\frac{17\cdots 57}{10\cdots 12}a^{6}-\frac{74\cdots 21}{10\cdots 12}a^{5}+\frac{78\cdots 42}{269894553023353}a^{4}+\frac{75\cdots 65}{539789106046706}a^{3}-\frac{53\cdots 47}{269894553023353}a^{2}-\frac{19\cdots 70}{269894553023353}a+\frac{23\cdots 93}{269894553023353}$, $\frac{46\cdots 49}{10\cdots 12}a^{20}-\frac{69\cdots 87}{539789106046706}a^{19}-\frac{33\cdots 11}{269894553023353}a^{18}+\frac{37\cdots 87}{10\cdots 12}a^{17}+\frac{82\cdots 85}{539789106046706}a^{16}-\frac{38\cdots 11}{10\cdots 12}a^{15}-\frac{56\cdots 49}{539789106046706}a^{14}+\frac{44\cdots 71}{25106470048684}a^{13}+\frac{11\cdots 20}{269894553023353}a^{12}-\frac{46\cdots 15}{10\cdots 12}a^{11}-\frac{59\cdots 99}{539789106046706}a^{10}+\frac{40\cdots 87}{10\cdots 12}a^{9}+\frac{81\cdots 63}{539789106046706}a^{8}+\frac{21\cdots 99}{10\cdots 12}a^{7}-\frac{88\cdots 47}{10\cdots 12}a^{6}-\frac{30\cdots 93}{10\cdots 12}a^{5}+\frac{40\cdots 04}{269894553023353}a^{4}+\frac{31\cdots 85}{539789106046706}a^{3}-\frac{58\cdots 21}{539789106046706}a^{2}-\frac{82\cdots 99}{269894553023353}a+\frac{10\cdots 15}{269894553023353}$, $\frac{47\cdots 25}{539789106046706}a^{20}-\frac{12\cdots 59}{539789106046706}a^{19}-\frac{71\cdots 58}{269894553023353}a^{18}+\frac{34\cdots 83}{539789106046706}a^{17}+\frac{89\cdots 80}{269894553023353}a^{16}-\frac{17\cdots 79}{269894553023353}a^{15}-\frac{63\cdots 17}{269894553023353}a^{14}+\frac{20\cdots 10}{6276617512171}a^{13}+\frac{27\cdots 67}{269894553023353}a^{12}-\frac{20\cdots 51}{269894553023353}a^{11}-\frac{68\cdots 80}{269894553023353}a^{10}+\frac{12\cdots 78}{269894553023353}a^{9}+\frac{93\cdots 68}{269894553023353}a^{8}+\frac{21\cdots 94}{269894553023353}a^{7}-\frac{99\cdots 27}{539789106046706}a^{6}-\frac{42\cdots 79}{539789106046706}a^{5}+\frac{87\cdots 29}{269894553023353}a^{4}+\frac{85\cdots 63}{539789106046706}a^{3}-\frac{59\cdots 17}{269894553023353}a^{2}-\frac{21\cdots 99}{269894553023353}a+\frac{25\cdots 65}{269894553023353}$, $\frac{65\cdots 21}{539789106046706}a^{20}-\frac{17\cdots 07}{539789106046706}a^{19}-\frac{19\cdots 55}{539789106046706}a^{18}+\frac{23\cdots 68}{269894553023353}a^{17}+\frac{24\cdots 11}{539789106046706}a^{16}-\frac{48\cdots 65}{539789106046706}a^{15}-\frac{17\cdots 27}{539789106046706}a^{14}+\frac{55\cdots 43}{12553235024342}a^{13}+\frac{74\cdots 17}{539789106046706}a^{12}-\frac{55\cdots 21}{539789106046706}a^{11}-\frac{19\cdots 67}{539789106046706}a^{10}+\frac{34\cdots 71}{539789106046706}a^{9}+\frac{25\cdots 89}{539789106046706}a^{8}+\frac{60\cdots 75}{539789106046706}a^{7}-\frac{68\cdots 83}{269894553023353}a^{6}-\frac{29\cdots 48}{269894553023353}a^{5}+\frac{12\cdots 15}{269894553023353}a^{4}+\frac{11\cdots 29}{539789106046706}a^{3}-\frac{82\cdots 53}{269894553023353}a^{2}-\frac{30\cdots 72}{269894553023353}a+\frac{36\cdots 69}{269894553023353}$, $\frac{36\cdots 77}{539789106046706}a^{20}-\frac{20\cdots 21}{10\cdots 12}a^{19}-\frac{54\cdots 68}{269894553023353}a^{18}+\frac{13\cdots 62}{269894553023353}a^{17}+\frac{27\cdots 01}{10\cdots 12}a^{16}-\frac{27\cdots 91}{539789106046706}a^{15}-\frac{19\cdots 53}{10\cdots 12}a^{14}+\frac{15\cdots 94}{6276617512171}a^{13}+\frac{82\cdots 91}{10\cdots 12}a^{12}-\frac{32\cdots 31}{539789106046706}a^{11}-\frac{21\cdots 13}{10\cdots 12}a^{10}+\frac{10\cdots 35}{269894553023353}a^{9}+\frac{28\cdots 25}{10\cdots 12}a^{8}+\frac{15\cdots 20}{269894553023353}a^{7}-\frac{15\cdots 05}{10\cdots 12}a^{6}-\frac{63\cdots 71}{10\cdots 12}a^{5}+\frac{26\cdots 03}{10\cdots 12}a^{4}+\frac{63\cdots 35}{539789106046706}a^{3}-\frac{91\cdots 31}{539789106046706}a^{2}-\frac{16\cdots 62}{269894553023353}a+\frac{19\cdots 55}{269894553023353}$, $\frac{19\cdots 63}{10\cdots 12}a^{20}-\frac{53\cdots 29}{10\cdots 12}a^{19}-\frac{14\cdots 64}{269894553023353}a^{18}+\frac{14\cdots 69}{10\cdots 12}a^{17}+\frac{72\cdots 21}{10\cdots 12}a^{16}-\frac{14\cdots 37}{10\cdots 12}a^{15}-\frac{51\cdots 21}{10\cdots 12}a^{14}+\frac{16\cdots 19}{25106470048684}a^{13}+\frac{21\cdots 53}{10\cdots 12}a^{12}-\frac{16\cdots 17}{10\cdots 12}a^{11}-\frac{55\cdots 77}{10\cdots 12}a^{10}+\frac{10\cdots 83}{10\cdots 12}a^{9}+\frac{75\cdots 81}{10\cdots 12}a^{8}+\frac{17\cdots 31}{10\cdots 12}a^{7}-\frac{20\cdots 19}{539789106046706}a^{6}-\frac{85\cdots 45}{539789106046706}a^{5}+\frac{70\cdots 05}{10\cdots 12}a^{4}+\frac{17\cdots 35}{539789106046706}a^{3}-\frac{11\cdots 30}{269894553023353}a^{2}-\frac{43\cdots 28}{269894553023353}a+\frac{50\cdots 16}{269894553023353}$, $\frac{24\cdots 65}{10\cdots 12}a^{20}-\frac{67\cdots 67}{10\cdots 12}a^{19}-\frac{37\cdots 93}{539789106046706}a^{18}+\frac{18\cdots 37}{10\cdots 12}a^{17}+\frac{94\cdots 07}{10\cdots 12}a^{16}-\frac{18\cdots 33}{10\cdots 12}a^{15}-\frac{66\cdots 91}{10\cdots 12}a^{14}+\frac{21\cdots 35}{25106470048684}a^{13}+\frac{28\cdots 43}{10\cdots 12}a^{12}-\frac{21\cdots 89}{10\cdots 12}a^{11}-\frac{72\cdots 31}{10\cdots 12}a^{10}+\frac{13\cdots 79}{10\cdots 12}a^{9}+\frac{98\cdots 47}{10\cdots 12}a^{8}+\frac{22\cdots 19}{10\cdots 12}a^{7}-\frac{26\cdots 91}{539789106046706}a^{6}-\frac{55\cdots 68}{269894553023353}a^{5}+\frac{93\cdots 29}{10\cdots 12}a^{4}+\frac{22\cdots 53}{539789106046706}a^{3}-\frac{16\cdots 88}{269894553023353}a^{2}-\frac{57\cdots 22}{269894553023353}a+\frac{69\cdots 16}{269894553023353}$, $\frac{34\cdots 67}{10\cdots 12}a^{20}-\frac{23\cdots 36}{269894553023353}a^{19}-\frac{25\cdots 50}{269894553023353}a^{18}+\frac{25\cdots 35}{10\cdots 12}a^{17}+\frac{64\cdots 83}{539789106046706}a^{16}-\frac{25\cdots 27}{10\cdots 12}a^{15}-\frac{45\cdots 51}{539789106046706}a^{14}+\frac{29\cdots 27}{25106470048684}a^{13}+\frac{96\cdots 69}{269894553023353}a^{12}-\frac{29\cdots 51}{10\cdots 12}a^{11}-\frac{49\cdots 17}{539789106046706}a^{10}+\frac{19\cdots 19}{10\cdots 12}a^{9}+\frac{66\cdots 77}{539789106046706}a^{8}+\frac{29\cdots 27}{10\cdots 12}a^{7}-\frac{71\cdots 49}{10\cdots 12}a^{6}-\frac{29\cdots 43}{10\cdots 12}a^{5}+\frac{31\cdots 38}{269894553023353}a^{4}+\frac{29\cdots 91}{539789106046706}a^{3}-\frac{43\cdots 37}{539789106046706}a^{2}-\frac{74\cdots 21}{269894553023353}a+\frac{89\cdots 85}{269894553023353}$, $\frac{11\cdots 47}{10\cdots 12}a^{20}-\frac{32\cdots 13}{10\cdots 12}a^{19}-\frac{17\cdots 97}{539789106046706}a^{18}+\frac{86\cdots 13}{10\cdots 12}a^{17}+\frac{43\cdots 51}{10\cdots 12}a^{16}-\frac{88\cdots 63}{10\cdots 12}a^{15}-\frac{30\cdots 79}{10\cdots 12}a^{14}+\frac{10\cdots 37}{25106470048684}a^{13}+\frac{13\cdots 59}{10\cdots 12}a^{12}-\frac{10\cdots 43}{10\cdots 12}a^{11}-\frac{33\cdots 15}{10\cdots 12}a^{10}+\frac{66\cdots 41}{10\cdots 12}a^{9}+\frac{44\cdots 55}{10\cdots 12}a^{8}+\frac{10\cdots 01}{10\cdots 12}a^{7}-\frac{59\cdots 20}{269894553023353}a^{6}-\frac{25\cdots 91}{269894553023353}a^{5}+\frac{41\cdots 25}{10\cdots 12}a^{4}+\frac{51\cdots 89}{269894553023353}a^{3}-\frac{67\cdots 39}{269894553023353}a^{2}-\frac{26\cdots 93}{269894553023353}a+\frac{30\cdots 16}{269894553023353}$, $\frac{22\cdots 05}{10\cdots 12}a^{20}-\frac{30\cdots 59}{539789106046706}a^{19}-\frac{16\cdots 63}{269894553023353}a^{18}+\frac{16\cdots 21}{10\cdots 12}a^{17}+\frac{41\cdots 37}{539789106046706}a^{16}-\frac{16\cdots 33}{10\cdots 12}a^{15}-\frac{29\cdots 91}{539789106046706}a^{14}+\frac{19\cdots 21}{25106470048684}a^{13}+\frac{63\cdots 68}{269894553023353}a^{12}-\frac{19\cdots 89}{10\cdots 12}a^{11}-\frac{32\cdots 35}{539789106046706}a^{10}+\frac{12\cdots 37}{10\cdots 12}a^{9}+\frac{43\cdots 81}{539789106046706}a^{8}+\frac{20\cdots 97}{10\cdots 12}a^{7}-\frac{46\cdots 43}{10\cdots 12}a^{6}-\frac{19\cdots 99}{10\cdots 12}a^{5}+\frac{20\cdots 65}{269894553023353}a^{4}+\frac{19\cdots 87}{539789106046706}a^{3}-\frac{28\cdots 77}{539789106046706}a^{2}-\frac{51\cdots 39}{269894553023353}a+\frac{61\cdots 57}{269894553023353}$
|
| |
| Regulator: | \( 12111978767.0 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 12111978767.0 \cdot 1}{2\cdot\sqrt{2406787169604002863343075235725312}}\cr\approx \mathstrut & 0.258878611311 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.7.36622433792.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 42 |
| Degree 7 sibling: | 7.7.36622433792.1 |
| Degree 14 sibling: | deg 14 |
| Minimal sibling: | 7.7.36622433792.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.7.0.1}{7} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.1.0a1.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 7.2.3.4a1.2 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
|
\(31\)
| 31.3.1.0a1.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 31.3.2.3a1.1 | $x^{6} + 2 x^{4} + 56 x^{3} + x^{2} + 87 x + 784$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 31.3.2.3a1.1 | $x^{6} + 2 x^{4} + 56 x^{3} + x^{2} + 87 x + 784$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 31.3.2.3a1.1 | $x^{6} + 2 x^{4} + 56 x^{3} + x^{2} + 87 x + 784$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |