Normalized defining polynomial
\( x^{28} - x^{27} + 18 x^{26} - 23 x^{25} + 29 x^{24} - 149 x^{23} - 795 x^{22} - 8 x^{21} - 2927 x^{20} + 667 x^{19} + 13846 x^{18} - 647 x^{17} + 109320 x^{16} + 98963 x^{15} + \cdots + 1148111 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(50201655190081835380839261671426578388690948486328125\) \(\medspace = 5^{21}\cdot 29^{26}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(76.24\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $5^{3/4}29^{13/14}\approx 76.23747453486138$ | ||
Ramified primes: | \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(145=5\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(67,·)$, $\chi_{145}(136,·)$, $\chi_{145}(138,·)$, $\chi_{145}(139,·)$, $\chi_{145}(13,·)$, $\chi_{145}(141,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(22,·)$, $\chi_{145}(24,·)$, $\chi_{145}(28,·)$, $\chi_{145}(93,·)$, $\chi_{145}(94,·)$, $\chi_{145}(33,·)$, $\chi_{145}(36,·)$, $\chi_{145}(38,·)$, $\chi_{145}(92,·)$, $\chi_{145}(42,·)$, $\chi_{145}(111,·)$, $\chi_{145}(49,·)$, $\chi_{145}(54,·)$, $\chi_{145}(57,·)$, $\chi_{145}(122,·)$, $\chi_{145}(59,·)$, $\chi_{145}(74,·)$, $\chi_{145}(62,·)$, $\chi_{145}(63,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{8192}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{1035241}a^{26}+\frac{167675}{1035241}a^{25}-\frac{8607}{1035241}a^{24}-\frac{49857}{1035241}a^{23}-\frac{327836}{1035241}a^{22}-\frac{226077}{1035241}a^{21}+\frac{22454}{1035241}a^{20}+\frac{89303}{1035241}a^{19}-\frac{285793}{1035241}a^{18}+\frac{14038}{1035241}a^{17}-\frac{69319}{1035241}a^{16}-\frac{323357}{1035241}a^{15}+\frac{64696}{1035241}a^{14}+\frac{455141}{1035241}a^{13}-\frac{285930}{1035241}a^{12}-\frac{332747}{1035241}a^{11}+\frac{504087}{1035241}a^{10}+\frac{50072}{1035241}a^{9}-\frac{71359}{1035241}a^{8}-\frac{479530}{1035241}a^{7}+\frac{431797}{1035241}a^{6}+\frac{22510}{1035241}a^{5}+\frac{37288}{1035241}a^{4}+\frac{214708}{1035241}a^{3}-\frac{58931}{1035241}a^{2}-\frac{343309}{1035241}a-\frac{485626}{1035241}$, $\frac{1}{26\!\cdots\!41}a^{27}-\frac{71\!\cdots\!17}{26\!\cdots\!41}a^{26}-\frac{11\!\cdots\!44}{26\!\cdots\!41}a^{25}-\frac{54\!\cdots\!88}{26\!\cdots\!41}a^{24}-\frac{87\!\cdots\!83}{26\!\cdots\!41}a^{23}-\frac{84\!\cdots\!48}{26\!\cdots\!41}a^{22}-\frac{57\!\cdots\!27}{26\!\cdots\!41}a^{21}+\frac{10\!\cdots\!26}{26\!\cdots\!41}a^{20}+\frac{11\!\cdots\!26}{26\!\cdots\!41}a^{19}+\frac{66\!\cdots\!56}{26\!\cdots\!41}a^{18}+\frac{11\!\cdots\!99}{26\!\cdots\!41}a^{17}+\frac{11\!\cdots\!63}{26\!\cdots\!41}a^{16}-\frac{52\!\cdots\!41}{26\!\cdots\!41}a^{15}+\frac{11\!\cdots\!21}{26\!\cdots\!41}a^{14}-\frac{61\!\cdots\!98}{26\!\cdots\!41}a^{13}+\frac{91\!\cdots\!19}{26\!\cdots\!41}a^{12}-\frac{84\!\cdots\!48}{26\!\cdots\!41}a^{11}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{10}-\frac{99\!\cdots\!42}{26\!\cdots\!41}a^{9}+\frac{59\!\cdots\!48}{26\!\cdots\!41}a^{8}+\frac{86\!\cdots\!70}{26\!\cdots\!41}a^{7}+\frac{12\!\cdots\!48}{26\!\cdots\!41}a^{6}+\frac{10\!\cdots\!12}{26\!\cdots\!41}a^{5}-\frac{12\!\cdots\!98}{26\!\cdots\!41}a^{4}+\frac{44\!\cdots\!03}{26\!\cdots\!41}a^{3}+\frac{81\!\cdots\!77}{26\!\cdots\!41}a^{2}-\frac{12\!\cdots\!37}{26\!\cdots\!41}a+\frac{10\!\cdots\!31}{26\!\cdots\!41}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{562}$, which has order $8992$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{42\!\cdots\!42}{15\!\cdots\!61}a^{27}-\frac{24\!\cdots\!90}{15\!\cdots\!61}a^{26}+\frac{75\!\cdots\!76}{15\!\cdots\!61}a^{25}-\frac{63\!\cdots\!12}{15\!\cdots\!61}a^{24}+\frac{10\!\cdots\!99}{15\!\cdots\!61}a^{23}-\frac{56\!\cdots\!85}{15\!\cdots\!61}a^{22}-\frac{36\!\cdots\!26}{15\!\cdots\!61}a^{21}-\frac{16\!\cdots\!24}{15\!\cdots\!61}a^{20}-\frac{13\!\cdots\!19}{15\!\cdots\!61}a^{19}-\frac{49\!\cdots\!68}{15\!\cdots\!61}a^{18}+\frac{55\!\cdots\!12}{15\!\cdots\!61}a^{17}+\frac{19\!\cdots\!95}{15\!\cdots\!61}a^{16}+\frac{47\!\cdots\!68}{15\!\cdots\!61}a^{15}+\frac{66\!\cdots\!97}{15\!\cdots\!61}a^{14}+\frac{16\!\cdots\!14}{15\!\cdots\!61}a^{13}+\frac{30\!\cdots\!33}{15\!\cdots\!61}a^{12}+\frac{50\!\cdots\!09}{15\!\cdots\!61}a^{11}+\frac{38\!\cdots\!27}{15\!\cdots\!61}a^{10}+\frac{10\!\cdots\!88}{15\!\cdots\!61}a^{9}+\frac{37\!\cdots\!04}{15\!\cdots\!61}a^{8}+\frac{51\!\cdots\!52}{15\!\cdots\!61}a^{7}+\frac{85\!\cdots\!64}{15\!\cdots\!61}a^{6}-\frac{19\!\cdots\!20}{15\!\cdots\!61}a^{5}-\frac{36\!\cdots\!67}{15\!\cdots\!61}a^{4}+\frac{56\!\cdots\!43}{15\!\cdots\!61}a^{3}-\frac{16\!\cdots\!67}{15\!\cdots\!61}a^{2}-\frac{12\!\cdots\!44}{15\!\cdots\!61}a+\frac{40\!\cdots\!07}{15\!\cdots\!61}$, $\frac{62\!\cdots\!60}{26\!\cdots\!41}a^{27}-\frac{44\!\cdots\!95}{26\!\cdots\!41}a^{26}+\frac{19\!\cdots\!05}{26\!\cdots\!41}a^{25}-\frac{96\!\cdots\!85}{26\!\cdots\!41}a^{24}+\frac{36\!\cdots\!55}{26\!\cdots\!41}a^{23}-\frac{50\!\cdots\!50}{26\!\cdots\!41}a^{22}+\frac{11\!\cdots\!95}{26\!\cdots\!41}a^{21}+\frac{11\!\cdots\!80}{26\!\cdots\!41}a^{20}-\frac{10\!\cdots\!70}{26\!\cdots\!41}a^{19}+\frac{13\!\cdots\!10}{26\!\cdots\!41}a^{18}-\frac{46\!\cdots\!20}{26\!\cdots\!41}a^{17}-\frac{43\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{19\!\cdots\!60}{26\!\cdots\!41}a^{15}-\frac{48\!\cdots\!45}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!70}{26\!\cdots\!41}a^{13}-\frac{18\!\cdots\!10}{26\!\cdots\!41}a^{12}+\frac{26\!\cdots\!25}{26\!\cdots\!41}a^{11}+\frac{30\!\cdots\!70}{26\!\cdots\!41}a^{10}+\frac{97\!\cdots\!35}{26\!\cdots\!41}a^{9}-\frac{44\!\cdots\!45}{26\!\cdots\!41}a^{8}+\frac{28\!\cdots\!40}{26\!\cdots\!41}a^{7}-\frac{18\!\cdots\!50}{26\!\cdots\!41}a^{6}+\frac{99\!\cdots\!31}{26\!\cdots\!41}a^{5}+\frac{49\!\cdots\!30}{26\!\cdots\!41}a^{4}-\frac{32\!\cdots\!00}{26\!\cdots\!41}a^{3}+\frac{53\!\cdots\!75}{26\!\cdots\!41}a^{2}+\frac{10\!\cdots\!30}{26\!\cdots\!41}a-\frac{81\!\cdots\!00}{26\!\cdots\!41}$, $\frac{65\!\cdots\!32}{26\!\cdots\!41}a^{27}-\frac{10\!\cdots\!40}{26\!\cdots\!41}a^{26}+\frac{11\!\cdots\!15}{26\!\cdots\!41}a^{25}-\frac{22\!\cdots\!97}{26\!\cdots\!41}a^{24}+\frac{21\!\cdots\!30}{26\!\cdots\!41}a^{23}-\frac{11\!\cdots\!95}{26\!\cdots\!41}a^{22}-\frac{44\!\cdots\!89}{26\!\cdots\!41}a^{21}+\frac{33\!\cdots\!20}{26\!\cdots\!41}a^{20}-\frac{14\!\cdots\!45}{26\!\cdots\!41}a^{19}+\frac{23\!\cdots\!86}{26\!\cdots\!41}a^{18}+\frac{97\!\cdots\!02}{26\!\cdots\!41}a^{17}-\frac{32\!\cdots\!85}{26\!\cdots\!41}a^{16}+\frac{63\!\cdots\!15}{26\!\cdots\!41}a^{15}+\frac{73\!\cdots\!86}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!51}{26\!\cdots\!41}a^{13}+\frac{45\!\cdots\!32}{26\!\cdots\!41}a^{12}+\frac{12\!\cdots\!89}{26\!\cdots\!41}a^{11}-\frac{60\!\cdots\!15}{26\!\cdots\!41}a^{10}+\frac{44\!\cdots\!78}{26\!\cdots\!41}a^{9}-\frac{16\!\cdots\!47}{26\!\cdots\!41}a^{8}+\frac{15\!\cdots\!26}{26\!\cdots\!41}a^{7}-\frac{61\!\cdots\!63}{26\!\cdots\!41}a^{6}-\frac{10\!\cdots\!57}{26\!\cdots\!41}a^{5}+\frac{60\!\cdots\!80}{26\!\cdots\!41}a^{4}+\frac{17\!\cdots\!43}{26\!\cdots\!41}a^{3}-\frac{55\!\cdots\!09}{26\!\cdots\!41}a^{2}-\frac{69\!\cdots\!87}{26\!\cdots\!41}a+\frac{24\!\cdots\!77}{26\!\cdots\!41}$, $\frac{16\!\cdots\!97}{26\!\cdots\!41}a^{27}-\frac{60\!\cdots\!70}{26\!\cdots\!41}a^{26}+\frac{38\!\cdots\!33}{26\!\cdots\!41}a^{25}-\frac{11\!\cdots\!37}{26\!\cdots\!41}a^{24}+\frac{22\!\cdots\!10}{26\!\cdots\!41}a^{23}-\frac{44\!\cdots\!50}{26\!\cdots\!41}a^{22}-\frac{48\!\cdots\!04}{26\!\cdots\!41}a^{21}+\frac{27\!\cdots\!52}{26\!\cdots\!41}a^{20}-\frac{79\!\cdots\!85}{26\!\cdots\!41}a^{19}+\frac{11\!\cdots\!36}{26\!\cdots\!41}a^{18}+\frac{40\!\cdots\!02}{26\!\cdots\!41}a^{17}-\frac{58\!\cdots\!65}{26\!\cdots\!41}a^{16}+\frac{22\!\cdots\!00}{26\!\cdots\!41}a^{15}-\frac{27\!\cdots\!49}{26\!\cdots\!41}a^{14}+\frac{62\!\cdots\!16}{26\!\cdots\!41}a^{13}-\frac{66\!\cdots\!83}{26\!\cdots\!41}a^{12}+\frac{12\!\cdots\!49}{26\!\cdots\!41}a^{11}-\frac{36\!\cdots\!43}{26\!\cdots\!41}a^{10}+\frac{67\!\cdots\!83}{26\!\cdots\!41}a^{9}-\frac{51\!\cdots\!92}{26\!\cdots\!41}a^{8}+\frac{12\!\cdots\!66}{26\!\cdots\!41}a^{7}-\frac{50\!\cdots\!98}{26\!\cdots\!41}a^{6}+\frac{26\!\cdots\!10}{26\!\cdots\!41}a^{5}-\frac{23\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{84\!\cdots\!87}{26\!\cdots\!41}a^{3}+\frac{26\!\cdots\!31}{26\!\cdots\!41}a^{2}+\frac{36\!\cdots\!98}{26\!\cdots\!41}a+\frac{87\!\cdots\!24}{26\!\cdots\!41}$, $\frac{62\!\cdots\!78}{26\!\cdots\!41}a^{27}+\frac{89\!\cdots\!75}{26\!\cdots\!41}a^{26}+\frac{15\!\cdots\!73}{26\!\cdots\!41}a^{25}-\frac{10\!\cdots\!28}{26\!\cdots\!41}a^{24}+\frac{87\!\cdots\!25}{26\!\cdots\!41}a^{23}-\frac{32\!\cdots\!60}{26\!\cdots\!41}a^{22}-\frac{54\!\cdots\!46}{26\!\cdots\!41}a^{21}-\frac{21\!\cdots\!59}{26\!\cdots\!41}a^{20}-\frac{55\!\cdots\!65}{26\!\cdots\!41}a^{19}+\frac{41\!\cdots\!94}{26\!\cdots\!41}a^{18}+\frac{15\!\cdots\!33}{26\!\cdots\!41}a^{17}+\frac{49\!\cdots\!75}{26\!\cdots\!41}a^{16}+\frac{16\!\cdots\!33}{26\!\cdots\!41}a^{15}+\frac{90\!\cdots\!14}{26\!\cdots\!41}a^{14}+\frac{80\!\cdots\!04}{26\!\cdots\!41}a^{13}+\frac{43\!\cdots\!53}{26\!\cdots\!41}a^{12}+\frac{11\!\cdots\!01}{26\!\cdots\!41}a^{11}+\frac{82\!\cdots\!65}{26\!\cdots\!41}a^{10}-\frac{23\!\cdots\!48}{26\!\cdots\!41}a^{9}-\frac{58\!\cdots\!13}{26\!\cdots\!41}a^{8}+\frac{51\!\cdots\!24}{26\!\cdots\!41}a^{7}-\frac{11\!\cdots\!02}{26\!\cdots\!41}a^{6}+\frac{61\!\cdots\!89}{26\!\cdots\!41}a^{5}+\frac{13\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{77\!\cdots\!58}{26\!\cdots\!41}a^{3}-\frac{30\!\cdots\!31}{26\!\cdots\!41}a^{2}-\frac{56\!\cdots\!78}{26\!\cdots\!41}a-\frac{37\!\cdots\!09}{26\!\cdots\!41}$, $\frac{84\!\cdots\!24}{26\!\cdots\!41}a^{27}-\frac{23\!\cdots\!70}{26\!\cdots\!41}a^{26}+\frac{16\!\cdots\!24}{26\!\cdots\!41}a^{25}-\frac{49\!\cdots\!84}{26\!\cdots\!41}a^{24}+\frac{70\!\cdots\!40}{26\!\cdots\!41}a^{23}-\frac{22\!\cdots\!25}{26\!\cdots\!41}a^{22}-\frac{16\!\cdots\!58}{26\!\cdots\!41}a^{21}+\frac{10\!\cdots\!59}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!65}{26\!\cdots\!41}a^{19}+\frac{60\!\cdots\!72}{26\!\cdots\!41}a^{18}-\frac{62\!\cdots\!31}{26\!\cdots\!41}a^{17}-\frac{20\!\cdots\!60}{26\!\cdots\!41}a^{16}+\frac{65\!\cdots\!27}{26\!\cdots\!41}a^{15}-\frac{13\!\cdots\!33}{26\!\cdots\!41}a^{14}+\frac{31\!\cdots\!02}{26\!\cdots\!41}a^{13}-\frac{22\!\cdots\!06}{26\!\cdots\!41}a^{12}+\frac{68\!\cdots\!08}{26\!\cdots\!41}a^{11}-\frac{13\!\cdots\!73}{26\!\cdots\!41}a^{10}+\frac{32\!\cdots\!06}{26\!\cdots\!41}a^{9}-\frac{12\!\cdots\!84}{26\!\cdots\!41}a^{8}+\frac{81\!\cdots\!62}{26\!\cdots\!41}a^{7}-\frac{38\!\cdots\!06}{26\!\cdots\!41}a^{6}+\frac{29\!\cdots\!53}{26\!\cdots\!41}a^{5}-\frac{16\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{56\!\cdots\!64}{26\!\cdots\!41}a^{3}+\frac{21\!\cdots\!02}{26\!\cdots\!41}a^{2}+\frac{26\!\cdots\!31}{26\!\cdots\!41}a-\frac{14\!\cdots\!34}{26\!\cdots\!41}$, $\frac{84\!\cdots\!73}{26\!\cdots\!41}a^{27}-\frac{37\!\cdots\!00}{26\!\cdots\!41}a^{26}+\frac{22\!\cdots\!09}{26\!\cdots\!41}a^{25}-\frac{70\!\cdots\!53}{26\!\cdots\!41}a^{24}+\frac{15\!\cdots\!70}{26\!\cdots\!41}a^{23}-\frac{22\!\cdots\!25}{26\!\cdots\!41}a^{22}-\frac{32\!\cdots\!46}{26\!\cdots\!41}a^{21}+\frac{16\!\cdots\!93}{26\!\cdots\!41}a^{20}-\frac{60\!\cdots\!20}{26\!\cdots\!41}a^{19}+\frac{56\!\cdots\!64}{26\!\cdots\!41}a^{18}+\frac{46\!\cdots\!33}{26\!\cdots\!41}a^{17}-\frac{38\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{15\!\cdots\!73}{26\!\cdots\!41}a^{15}-\frac{14\!\cdots\!16}{26\!\cdots\!41}a^{14}+\frac{31\!\cdots\!14}{26\!\cdots\!41}a^{13}+\frac{15\!\cdots\!23}{26\!\cdots\!41}a^{12}+\frac{55\!\cdots\!41}{26\!\cdots\!41}a^{11}-\frac{22\!\cdots\!70}{26\!\cdots\!41}a^{10}+\frac{35\!\cdots\!77}{26\!\cdots\!41}a^{9}-\frac{50\!\cdots\!08}{26\!\cdots\!41}a^{8}+\frac{38\!\cdots\!04}{26\!\cdots\!41}a^{7}-\frac{12\!\cdots\!92}{26\!\cdots\!41}a^{6}+\frac{23\!\cdots\!57}{26\!\cdots\!41}a^{5}-\frac{62\!\cdots\!00}{26\!\cdots\!41}a^{4}-\frac{28\!\cdots\!23}{26\!\cdots\!41}a^{3}+\frac{44\!\cdots\!29}{26\!\cdots\!41}a^{2}+\frac{10\!\cdots\!67}{26\!\cdots\!41}a-\frac{39\!\cdots\!83}{26\!\cdots\!41}$, $\frac{25\!\cdots\!29}{26\!\cdots\!41}a^{27}-\frac{86\!\cdots\!51}{26\!\cdots\!41}a^{26}+\frac{45\!\cdots\!13}{26\!\cdots\!41}a^{25}-\frac{29\!\cdots\!11}{26\!\cdots\!41}a^{24}+\frac{58\!\cdots\!41}{26\!\cdots\!41}a^{23}-\frac{35\!\cdots\!12}{26\!\cdots\!41}a^{22}-\frac{22\!\cdots\!63}{26\!\cdots\!41}a^{21}-\frac{15\!\cdots\!74}{26\!\cdots\!41}a^{20}-\frac{84\!\cdots\!77}{26\!\cdots\!41}a^{19}-\frac{34\!\cdots\!06}{26\!\cdots\!41}a^{18}+\frac{31\!\cdots\!27}{26\!\cdots\!41}a^{17}+\frac{21\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{29\!\cdots\!74}{26\!\cdots\!41}a^{15}+\frac{43\!\cdots\!17}{26\!\cdots\!41}a^{14}+\frac{11\!\cdots\!15}{26\!\cdots\!41}a^{13}+\frac{19\!\cdots\!91}{26\!\cdots\!41}a^{12}+\frac{32\!\cdots\!00}{26\!\cdots\!41}a^{11}+\frac{27\!\cdots\!38}{26\!\cdots\!41}a^{10}+\frac{65\!\cdots\!44}{26\!\cdots\!41}a^{9}+\frac{26\!\cdots\!63}{26\!\cdots\!41}a^{8}+\frac{43\!\cdots\!10}{26\!\cdots\!41}a^{7}+\frac{36\!\cdots\!04}{26\!\cdots\!41}a^{6}-\frac{57\!\cdots\!83}{26\!\cdots\!41}a^{5}-\frac{56\!\cdots\!35}{26\!\cdots\!41}a^{4}+\frac{15\!\cdots\!96}{26\!\cdots\!41}a^{3}-\frac{10\!\cdots\!62}{26\!\cdots\!41}a^{2}-\frac{74\!\cdots\!12}{26\!\cdots\!41}a-\frac{42\!\cdots\!95}{26\!\cdots\!41}$, $\frac{83\!\cdots\!17}{26\!\cdots\!41}a^{27}-\frac{13\!\cdots\!32}{26\!\cdots\!41}a^{26}+\frac{13\!\cdots\!93}{26\!\cdots\!41}a^{25}-\frac{33\!\cdots\!12}{26\!\cdots\!41}a^{24}-\frac{73\!\cdots\!78}{26\!\cdots\!41}a^{23}-\frac{78\!\cdots\!47}{26\!\cdots\!41}a^{22}-\frac{81\!\cdots\!24}{26\!\cdots\!41}a^{21}-\frac{54\!\cdots\!71}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!99}{26\!\cdots\!41}a^{19}-\frac{20\!\cdots\!17}{26\!\cdots\!41}a^{18}+\frac{14\!\cdots\!44}{26\!\cdots\!41}a^{17}+\frac{94\!\cdots\!71}{26\!\cdots\!41}a^{16}+\frac{81\!\cdots\!82}{26\!\cdots\!41}a^{15}+\frac{17\!\cdots\!07}{26\!\cdots\!41}a^{14}+\frac{26\!\cdots\!45}{26\!\cdots\!41}a^{13}+\frac{63\!\cdots\!77}{26\!\cdots\!41}a^{12}+\frac{87\!\cdots\!97}{26\!\cdots\!41}a^{11}+\frac{63\!\cdots\!41}{26\!\cdots\!41}a^{10}+\frac{14\!\cdots\!21}{26\!\cdots\!41}a^{9}+\frac{11\!\cdots\!11}{26\!\cdots\!41}a^{8}-\frac{82\!\cdots\!40}{26\!\cdots\!41}a^{7}+\frac{27\!\cdots\!23}{26\!\cdots\!41}a^{6}-\frac{10\!\cdots\!86}{26\!\cdots\!41}a^{5}-\frac{76\!\cdots\!46}{26\!\cdots\!41}a^{4}+\frac{32\!\cdots\!82}{26\!\cdots\!41}a^{3}-\frac{39\!\cdots\!26}{26\!\cdots\!41}a^{2}-\frac{32\!\cdots\!01}{26\!\cdots\!41}a+\frac{65\!\cdots\!25}{26\!\cdots\!41}$, $\frac{92\!\cdots\!78}{26\!\cdots\!41}a^{27}-\frac{10\!\cdots\!88}{26\!\cdots\!41}a^{26}+\frac{15\!\cdots\!17}{26\!\cdots\!41}a^{25}-\frac{20\!\cdots\!88}{26\!\cdots\!41}a^{24}+\frac{12\!\cdots\!25}{26\!\cdots\!41}a^{23}-\frac{83\!\cdots\!34}{26\!\cdots\!41}a^{22}-\frac{76\!\cdots\!32}{26\!\cdots\!41}a^{21}+\frac{20\!\cdots\!76}{26\!\cdots\!41}a^{20}-\frac{22\!\cdots\!76}{26\!\cdots\!41}a^{19}-\frac{10\!\cdots\!87}{26\!\cdots\!41}a^{18}+\frac{14\!\cdots\!91}{26\!\cdots\!41}a^{17}-\frac{77\!\cdots\!89}{26\!\cdots\!41}a^{16}+\frac{86\!\cdots\!99}{26\!\cdots\!41}a^{15}+\frac{11\!\cdots\!82}{26\!\cdots\!41}a^{14}+\frac{21\!\cdots\!55}{26\!\cdots\!41}a^{13}+\frac{55\!\cdots\!47}{26\!\cdots\!41}a^{12}+\frac{79\!\cdots\!35}{26\!\cdots\!41}a^{11}+\frac{45\!\cdots\!18}{26\!\cdots\!41}a^{10}+\frac{23\!\cdots\!20}{26\!\cdots\!41}a^{9}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{8}+\frac{27\!\cdots\!65}{26\!\cdots\!41}a^{7}+\frac{36\!\cdots\!84}{26\!\cdots\!41}a^{6}-\frac{15\!\cdots\!03}{26\!\cdots\!41}a^{5}-\frac{28\!\cdots\!68}{26\!\cdots\!41}a^{4}+\frac{30\!\cdots\!16}{26\!\cdots\!41}a^{3}-\frac{29\!\cdots\!81}{26\!\cdots\!41}a^{2}-\frac{25\!\cdots\!09}{26\!\cdots\!41}a+\frac{15\!\cdots\!62}{26\!\cdots\!41}$, $\frac{46\!\cdots\!10}{26\!\cdots\!41}a^{27}-\frac{19\!\cdots\!77}{26\!\cdots\!41}a^{26}+\frac{85\!\cdots\!56}{26\!\cdots\!41}a^{25}-\frac{63\!\cdots\!93}{26\!\cdots\!41}a^{24}+\frac{13\!\cdots\!16}{26\!\cdots\!41}a^{23}-\frac{71\!\cdots\!29}{26\!\cdots\!41}a^{22}-\frac{40\!\cdots\!19}{26\!\cdots\!41}a^{21}-\frac{27\!\cdots\!76}{26\!\cdots\!41}a^{20}-\frac{16\!\cdots\!17}{26\!\cdots\!41}a^{19}-\frac{44\!\cdots\!78}{26\!\cdots\!41}a^{18}+\frac{56\!\cdots\!68}{26\!\cdots\!41}a^{17}+\frac{38\!\cdots\!61}{26\!\cdots\!41}a^{16}+\frac{55\!\cdots\!47}{26\!\cdots\!41}a^{15}+\frac{75\!\cdots\!09}{26\!\cdots\!41}a^{14}+\frac{21\!\cdots\!99}{26\!\cdots\!41}a^{13}+\frac{34\!\cdots\!12}{26\!\cdots\!41}a^{12}+\frac{61\!\cdots\!96}{26\!\cdots\!41}a^{11}+\frac{50\!\cdots\!88}{26\!\cdots\!41}a^{10}+\frac{12\!\cdots\!79}{26\!\cdots\!41}a^{9}+\frac{35\!\cdots\!19}{26\!\cdots\!41}a^{8}+\frac{95\!\cdots\!95}{26\!\cdots\!41}a^{7}+\frac{46\!\cdots\!80}{26\!\cdots\!41}a^{6}-\frac{51\!\cdots\!86}{26\!\cdots\!41}a^{5}+\frac{30\!\cdots\!67}{26\!\cdots\!41}a^{4}+\frac{10\!\cdots\!21}{26\!\cdots\!41}a^{3}-\frac{19\!\cdots\!75}{26\!\cdots\!41}a^{2}-\frac{12\!\cdots\!33}{26\!\cdots\!41}a-\frac{37\!\cdots\!75}{26\!\cdots\!41}$, $\frac{35\!\cdots\!84}{26\!\cdots\!41}a^{27}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{26}+\frac{36\!\cdots\!99}{26\!\cdots\!41}a^{25}+\frac{20\!\cdots\!89}{26\!\cdots\!41}a^{24}-\frac{47\!\cdots\!26}{26\!\cdots\!41}a^{23}+\frac{16\!\cdots\!99}{26\!\cdots\!41}a^{22}-\frac{54\!\cdots\!62}{26\!\cdots\!41}a^{21}-\frac{99\!\cdots\!16}{26\!\cdots\!41}a^{20}-\frac{11\!\cdots\!38}{26\!\cdots\!41}a^{19}-\frac{39\!\cdots\!25}{26\!\cdots\!41}a^{18}+\frac{96\!\cdots\!33}{26\!\cdots\!41}a^{17}+\frac{19\!\cdots\!78}{26\!\cdots\!41}a^{16}+\frac{23\!\cdots\!09}{26\!\cdots\!41}a^{15}+\frac{19\!\cdots\!11}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!70}{26\!\cdots\!41}a^{13}+\frac{54\!\cdots\!69}{26\!\cdots\!41}a^{12}+\frac{55\!\cdots\!28}{26\!\cdots\!41}a^{11}+\frac{75\!\cdots\!40}{26\!\cdots\!41}a^{10}+\frac{11\!\cdots\!05}{26\!\cdots\!41}a^{9}+\frac{21\!\cdots\!78}{26\!\cdots\!41}a^{8}-\frac{23\!\cdots\!19}{26\!\cdots\!41}a^{7}+\frac{29\!\cdots\!32}{26\!\cdots\!41}a^{6}-\frac{11\!\cdots\!18}{26\!\cdots\!41}a^{5}-\frac{31\!\cdots\!59}{26\!\cdots\!41}a^{4}+\frac{33\!\cdots\!71}{26\!\cdots\!41}a^{3}-\frac{35\!\cdots\!44}{26\!\cdots\!41}a^{2}-\frac{30\!\cdots\!14}{26\!\cdots\!41}a+\frac{63\!\cdots\!21}{26\!\cdots\!41}$, $\frac{59\!\cdots\!56}{26\!\cdots\!41}a^{27}+\frac{37\!\cdots\!55}{26\!\cdots\!41}a^{26}+\frac{10\!\cdots\!16}{26\!\cdots\!41}a^{25}-\frac{19\!\cdots\!03}{26\!\cdots\!41}a^{24}+\frac{58\!\cdots\!91}{26\!\cdots\!41}a^{23}-\frac{65\!\cdots\!77}{26\!\cdots\!41}a^{22}-\frac{57\!\cdots\!24}{26\!\cdots\!41}a^{21}-\frac{54\!\cdots\!93}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!27}{26\!\cdots\!41}a^{19}-\frac{19\!\cdots\!08}{26\!\cdots\!41}a^{18}+\frac{80\!\cdots\!23}{26\!\cdots\!41}a^{17}+\frac{78\!\cdots\!23}{26\!\cdots\!41}a^{16}+\frac{66\!\cdots\!83}{26\!\cdots\!41}a^{15}+\frac{13\!\cdots\!58}{26\!\cdots\!41}a^{14}+\frac{27\!\cdots\!49}{26\!\cdots\!41}a^{13}+\frac{55\!\cdots\!31}{26\!\cdots\!41}a^{12}+\frac{94\!\cdots\!61}{26\!\cdots\!41}a^{11}+\frac{85\!\cdots\!96}{26\!\cdots\!41}a^{10}+\frac{17\!\cdots\!64}{26\!\cdots\!41}a^{9}+\frac{13\!\cdots\!43}{26\!\cdots\!41}a^{8}+\frac{38\!\cdots\!26}{26\!\cdots\!41}a^{7}+\frac{21\!\cdots\!02}{26\!\cdots\!41}a^{6}-\frac{46\!\cdots\!85}{26\!\cdots\!41}a^{5}-\frac{94\!\cdots\!92}{26\!\cdots\!41}a^{4}+\frac{12\!\cdots\!63}{26\!\cdots\!41}a^{3}-\frac{33\!\cdots\!93}{26\!\cdots\!41}a^{2}-\frac{24\!\cdots\!79}{26\!\cdots\!41}a-\frac{51\!\cdots\!13}{26\!\cdots\!41}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 34681517373.86067 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 8992}{2\cdot\sqrt{50201655190081835380839261671426578388690948486328125}}\cr\approx \mathstrut & 0.104012020207135 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 28 |
The 28 conjugacy class representatives for $C_{28}$ |
Character table for $C_{28}$ is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.0.105125.2, 7.7.594823321.1, 14.14.27641779937927268828125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $28$ | $28$ | R | $28$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | $28$ | ${\href{/padicField/17.4.0.1}{4} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }^{4}$ | $28$ | R | ${\href{/padicField/31.14.0.1}{14} }^{2}$ | $28$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | $28$ | $28$ | $28$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $28$ | $4$ | $7$ | $21$ | |||
\(29\) | 29.14.13.11 | $x^{14} + 348$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
29.14.13.11 | $x^{14} + 348$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |