Normalized defining polynomial

\( x^{20} + 20 x^{18} + 565 x^{16} + 29472 x^{14} + 869610 x^{12} + 15155944 x^{10} + 170472586 x^{8} + \cdots + 22163170129 \) x^20 + 20*x^18 + 565*x^16 + 29472*x^14 + 869610*x^12 + 15155944*x^10 + 170472586*x^8 + 1270626240*x^6 + 6120549669*x^4 + 17367927780*x^2 + 22163170129

Invariants

Degree:		$20$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 10]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(3525175555633378160676822382018560000000000\) 3525175555633378160676822382018560000000000 \(\medspace = 2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(134.08\)	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:		$2^{2}3^{1/2}5^{1/2}11^{9/10}\approx 134.0784675354655$
Ramified primes:		\(2\), \(3\), \(5\), \(11\) 2, 3, 5, 11	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$20$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(211,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(1051,·)$, $\chi_{1320}(479,·)$, $\chi_{1320}(359,·)$, $\chi_{1320}(1319,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(811,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(239,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(571,·)$, $\chi_{1320}(931,·)$, $\chi_{1320}(959,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{512}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{8}-\frac{1}{8}a^{4}-\frac{3}{16}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{5}-\frac{3}{16}a$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{3}{16}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{8}a^{7}-\frac{3}{16}a^{3}$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{8}-\frac{5}{64}a^{4}-\frac{3}{64}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{9}-\frac{5}{64}a^{5}-\frac{3}{64}a$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{10}-\frac{5}{64}a^{6}-\frac{3}{64}a^{2}$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{11}-\frac{5}{64}a^{7}-\frac{3}{64}a^{3}$, $\frac{1}{256}a^{16}-\frac{3}{128}a^{8}-\frac{1}{32}a^{4}-\frac{3}{256}$, $\frac{1}{256}a^{17}-\frac{3}{128}a^{9}-\frac{1}{32}a^{5}-\frac{3}{256}a$, $\frac{1}{18\!\cdots\!96}a^{18}+\frac{32\!\cdots\!57}{18\!\cdots\!96}a^{16}+\frac{37\!\cdots\!69}{22\!\cdots\!12}a^{14}+\frac{94\!\cdots\!15}{45\!\cdots\!24}a^{12}-\frac{16\!\cdots\!11}{91\!\cdots\!48}a^{10}-\frac{20\!\cdots\!29}{91\!\cdots\!48}a^{8}-\frac{35\!\cdots\!25}{11\!\cdots\!56}a^{6}+\frac{46\!\cdots\!63}{45\!\cdots\!24}a^{4}-\frac{59\!\cdots\!99}{18\!\cdots\!96}a^{2}-\frac{58\!\cdots\!71}{18\!\cdots\!96}$, $\frac{1}{27\!\cdots\!08}a^{19}+\frac{15\!\cdots\!89}{13\!\cdots\!04}a^{17}-\frac{37\!\cdots\!63}{68\!\cdots\!52}a^{15}-\frac{17\!\cdots\!61}{68\!\cdots\!52}a^{13}+\frac{39\!\cdots\!99}{13\!\cdots\!04}a^{11}+\frac{10\!\cdots\!05}{34\!\cdots\!76}a^{9}+\frac{14\!\cdots\!33}{68\!\cdots\!52}a^{7}-\frac{31\!\cdots\!83}{68\!\cdots\!52}a^{5}+\frac{54\!\cdots\!37}{27\!\cdots\!08}a^{3}-\frac{54\!\cdots\!61}{13\!\cdots\!04}a$ 1, a, a^2, a^3, 1/4*a^4 + 1/4, 1/4*a^5 + 1/4*a, 1/4*a^6 + 1/4*a^2, 1/4*a^7 + 1/4*a^3, 1/16*a^8 - 1/8*a^4 - 3/16, 1/16*a^9 - 1/8*a^5 - 3/16*a, 1/16*a^10 - 1/8*a^6 - 3/16*a^2, 1/16*a^11 - 1/8*a^7 - 3/16*a^3, 1/64*a^12 - 1/64*a^8 - 5/64*a^4 - 3/64, 1/64*a^13 - 1/64*a^9 - 5/64*a^5 - 3/64*a, 1/64*a^14 - 1/64*a^10 - 5/64*a^6 - 3/64*a^2, 1/64*a^15 - 1/64*a^11 - 5/64*a^7 - 3/64*a^3, 1/256*a^16 - 3/128*a^8 - 1/32*a^4 - 3/256, 1/256*a^17 - 3/128*a^9 - 1/32*a^5 - 3/256*a, 1/18368320056708035522301220096*a^18 + 3294822432575180717732357/18368320056708035522301220096*a^16 + 3776828420365165816257869/2296040007088504440287652512*a^14 + 9495797369074024367105015/4592080014177008880575305024*a^12 - 169773933600762703576537911/9184160028354017761150610048*a^10 - 205248569586050902647228429/9184160028354017761150610048*a^8 - 35308771851755250206199125/1148020003544252220143826256*a^6 + 467275185353200994784523363/4592080014177008880575305024*a^4 - 5970179305688270506660613499/18368320056708035522301220096*a^2 - 5841014831145036702541386371/18368320056708035522301220096, 1/2734546911802295372311549539351808*a^19 + 1574259952896382902344460114289/1367273455901147686155774769675904*a^17 - 3756959659191946175852797896163/683636727950573843077887384837952*a^15 - 1731778679549135400062594802161/683636727950573843077887384837952*a^13 + 39851192111598469586676582264899/1367273455901147686155774769675904*a^11 + 1058082312686851834369090802505/341818363975286921538943692418976*a^9 + 14695447576531901101769018139133/683636727950573843077887384837952*a^7 - 31247632703786091010627540316983/683636727950573843077887384837952*a^5 + 547111376004795427481482301383137/2734546911802295372311549539351808*a^3 - 542356665313685145956499064124261/1367273455901147686155774769675904*a

Class group and class number

$C_{2}\times C_{2}\times C_{1198220}$, which has order $4792880$ (assuming GRH)

Unit group

Rank:		$9$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -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:		$\frac{74\!\cdots\!97}{11\!\cdots\!56}a^{18}+\frac{24\!\cdots\!85}{22\!\cdots\!12}a^{16}+\frac{14\!\cdots\!01}{45\!\cdots\!24}a^{14}+\frac{41\!\cdots\!57}{22\!\cdots\!12}a^{12}+\frac{22\!\cdots\!55}{45\!\cdots\!24}a^{10}+\frac{17\!\cdots\!67}{22\!\cdots\!12}a^{8}+\frac{35\!\cdots\!99}{45\!\cdots\!24}a^{6}+\frac{10\!\cdots\!91}{22\!\cdots\!12}a^{4}+\frac{76\!\cdots\!29}{45\!\cdots\!24}a^{2}+\frac{15\!\cdots\!53}{57\!\cdots\!28}$, $\frac{36\!\cdots\!05}{91\!\cdots\!48}a^{18}+\frac{47\!\cdots\!70}{71\!\cdots\!41}a^{16}+\frac{89\!\cdots\!05}{45\!\cdots\!24}a^{14}+\frac{25\!\cdots\!83}{22\!\cdots\!12}a^{12}+\frac{35\!\cdots\!25}{11\!\cdots\!56}a^{10}+\frac{11\!\cdots\!25}{22\!\cdots\!12}a^{8}+\frac{21\!\cdots\!67}{45\!\cdots\!24}a^{6}+\frac{66\!\cdots\!65}{22\!\cdots\!12}a^{4}+\frac{94\!\cdots\!19}{91\!\cdots\!48}a^{2}+\frac{37\!\cdots\!99}{22\!\cdots\!12}$, $\frac{93\!\cdots\!01}{18\!\cdots\!96}a^{18}+\frac{97\!\cdots\!65}{91\!\cdots\!48}a^{16}+\frac{67\!\cdots\!55}{22\!\cdots\!12}a^{14}+\frac{69\!\cdots\!65}{45\!\cdots\!24}a^{12}+\frac{41\!\cdots\!61}{91\!\cdots\!48}a^{10}+\frac{91\!\cdots\!67}{11\!\cdots\!56}a^{8}+\frac{51\!\cdots\!51}{57\!\cdots\!28}a^{6}+\frac{29\!\cdots\!55}{45\!\cdots\!24}a^{4}+\frac{49\!\cdots\!97}{18\!\cdots\!96}a^{2}+\frac{47\!\cdots\!75}{91\!\cdots\!48}$, $\frac{93\!\cdots\!01}{18\!\cdots\!96}a^{18}+\frac{97\!\cdots\!65}{91\!\cdots\!48}a^{16}+\frac{67\!\cdots\!55}{22\!\cdots\!12}a^{14}+\frac{69\!\cdots\!65}{45\!\cdots\!24}a^{12}+\frac{41\!\cdots\!61}{91\!\cdots\!48}a^{10}+\frac{91\!\cdots\!67}{11\!\cdots\!56}a^{8}+\frac{51\!\cdots\!51}{57\!\cdots\!28}a^{6}+\frac{29\!\cdots\!55}{45\!\cdots\!24}a^{4}+\frac{49\!\cdots\!97}{18\!\cdots\!96}a^{2}+\frac{47\!\cdots\!23}{91\!\cdots\!48}$, $\frac{12\!\cdots\!33}{13\!\cdots\!04}a^{19}+\frac{23\!\cdots\!25}{45\!\cdots\!24}a^{18}+\frac{97\!\cdots\!11}{68\!\cdots\!52}a^{17}+\frac{13\!\cdots\!95}{18\!\cdots\!96}a^{16}+\frac{32\!\cdots\!49}{68\!\cdots\!52}a^{15}+\frac{10\!\cdots\!57}{45\!\cdots\!24}a^{14}+\frac{17\!\cdots\!47}{68\!\cdots\!52}a^{13}+\frac{63\!\cdots\!51}{45\!\cdots\!24}a^{12}+\frac{11\!\cdots\!53}{17\!\cdots\!88}a^{11}+\frac{16\!\cdots\!21}{45\!\cdots\!24}a^{10}+\frac{75\!\cdots\!03}{68\!\cdots\!52}a^{9}+\frac{47\!\cdots\!17}{91\!\cdots\!48}a^{8}+\frac{75\!\cdots\!27}{68\!\cdots\!52}a^{7}+\frac{20\!\cdots\!07}{45\!\cdots\!24}a^{6}+\frac{47\!\cdots\!73}{68\!\cdots\!52}a^{5}+\frac{11\!\cdots\!19}{45\!\cdots\!24}a^{4}+\frac{35\!\cdots\!55}{13\!\cdots\!04}a^{3}+\frac{15\!\cdots\!91}{22\!\cdots\!12}a^{2}+\frac{15\!\cdots\!75}{34\!\cdots\!76}a+\frac{16\!\cdots\!23}{18\!\cdots\!96}$, $\frac{17\!\cdots\!01}{34\!\cdots\!76}a^{19}-\frac{32\!\cdots\!13}{45\!\cdots\!24}a^{18}+\frac{76\!\cdots\!81}{13\!\cdots\!04}a^{17}-\frac{21\!\cdots\!75}{18\!\cdots\!96}a^{16}+\frac{16\!\cdots\!11}{68\!\cdots\!52}a^{15}-\frac{77\!\cdots\!29}{22\!\cdots\!12}a^{14}+\frac{90\!\cdots\!35}{68\!\cdots\!52}a^{13}-\frac{88\!\cdots\!65}{45\!\cdots\!24}a^{12}+\frac{22\!\cdots\!01}{68\!\cdots\!52}a^{11}-\frac{30\!\cdots\!97}{57\!\cdots\!28}a^{10}+\frac{16\!\cdots\!07}{34\!\cdots\!76}a^{9}-\frac{76\!\cdots\!85}{91\!\cdots\!48}a^{8}+\frac{29\!\cdots\!97}{68\!\cdots\!52}a^{7}-\frac{18\!\cdots\!03}{22\!\cdots\!12}a^{6}+\frac{16\!\cdots\!69}{68\!\cdots\!52}a^{5}-\frac{22\!\cdots\!01}{45\!\cdots\!24}a^{4}+\frac{52\!\cdots\!53}{68\!\cdots\!52}a^{3}-\frac{79\!\cdots\!11}{45\!\cdots\!24}a^{2}+\frac{14\!\cdots\!95}{13\!\cdots\!04}a-\frac{49\!\cdots\!23}{18\!\cdots\!96}$, $\frac{13\!\cdots\!63}{13\!\cdots\!04}a^{19}+\frac{13\!\cdots\!53}{18\!\cdots\!96}a^{18}+\frac{84\!\cdots\!87}{68\!\cdots\!52}a^{17}+\frac{22\!\cdots\!05}{18\!\cdots\!96}a^{16}+\frac{32\!\cdots\!25}{68\!\cdots\!52}a^{15}+\frac{26\!\cdots\!87}{71\!\cdots\!41}a^{14}+\frac{17\!\cdots\!43}{68\!\cdots\!52}a^{13}+\frac{47\!\cdots\!15}{22\!\cdots\!12}a^{12}+\frac{23\!\cdots\!67}{34\!\cdots\!76}a^{11}+\frac{53\!\cdots\!13}{91\!\cdots\!48}a^{10}+\frac{68\!\cdots\!23}{68\!\cdots\!52}a^{9}+\frac{83\!\cdots\!21}{91\!\cdots\!48}a^{8}+\frac{63\!\cdots\!43}{68\!\cdots\!52}a^{7}+\frac{20\!\cdots\!07}{22\!\cdots\!12}a^{6}+\frac{37\!\cdots\!21}{68\!\cdots\!52}a^{5}+\frac{31\!\cdots\!07}{57\!\cdots\!28}a^{4}+\frac{25\!\cdots\!69}{13\!\cdots\!04}a^{3}+\frac{36\!\cdots\!41}{18\!\cdots\!96}a^{2}+\frac{95\!\cdots\!25}{34\!\cdots\!76}a+\frac{59\!\cdots\!65}{18\!\cdots\!96}$, $\frac{12\!\cdots\!73}{27\!\cdots\!08}a^{19}-\frac{20\!\cdots\!63}{18\!\cdots\!96}a^{18}+\frac{17\!\cdots\!51}{27\!\cdots\!08}a^{17}-\frac{16\!\cdots\!65}{91\!\cdots\!48}a^{16}+\frac{18\!\cdots\!71}{85\!\cdots\!44}a^{15}-\frac{15\!\cdots\!51}{28\!\cdots\!64}a^{14}+\frac{10\!\cdots\!97}{85\!\cdots\!44}a^{13}-\frac{14\!\cdots\!45}{45\!\cdots\!24}a^{12}+\frac{44\!\cdots\!61}{13\!\cdots\!04}a^{11}-\frac{77\!\cdots\!71}{91\!\cdots\!48}a^{10}+\frac{67\!\cdots\!43}{13\!\cdots\!04}a^{9}-\frac{15\!\cdots\!13}{11\!\cdots\!56}a^{8}+\frac{16\!\cdots\!79}{34\!\cdots\!76}a^{7}-\frac{30\!\cdots\!37}{22\!\cdots\!12}a^{6}+\frac{98\!\cdots\!33}{34\!\cdots\!76}a^{5}-\frac{37\!\cdots\!67}{45\!\cdots\!24}a^{4}+\frac{27\!\cdots\!45}{27\!\cdots\!08}a^{3}-\frac{54\!\cdots\!51}{18\!\cdots\!96}a^{2}+\frac{44\!\cdots\!35}{27\!\cdots\!08}a-\frac{43\!\cdots\!67}{91\!\cdots\!48}$, $\frac{10\!\cdots\!07}{27\!\cdots\!08}a^{19}-\frac{10\!\cdots\!51}{18\!\cdots\!96}a^{18}+\frac{13\!\cdots\!07}{27\!\cdots\!08}a^{17}-\frac{16\!\cdots\!55}{18\!\cdots\!96}a^{16}+\frac{11\!\cdots\!19}{68\!\cdots\!52}a^{15}-\frac{60\!\cdots\!17}{22\!\cdots\!12}a^{14}+\frac{65\!\cdots\!29}{68\!\cdots\!52}a^{13}-\frac{34\!\cdots\!99}{22\!\cdots\!12}a^{12}+\frac{34\!\cdots\!37}{13\!\cdots\!04}a^{11}-\frac{38\!\cdots\!07}{91\!\cdots\!48}a^{10}+\frac{52\!\cdots\!37}{13\!\cdots\!04}a^{9}-\frac{59\!\cdots\!15}{91\!\cdots\!48}a^{8}+\frac{25\!\cdots\!31}{68\!\cdots\!52}a^{7}-\frac{36\!\cdots\!73}{57\!\cdots\!28}a^{6}+\frac{15\!\cdots\!93}{68\!\cdots\!52}a^{5}-\frac{43\!\cdots\!77}{11\!\cdots\!56}a^{4}+\frac{20\!\cdots\!35}{27\!\cdots\!08}a^{3}-\frac{24\!\cdots\!91}{18\!\cdots\!96}a^{2}+\frac{31\!\cdots\!35}{27\!\cdots\!08}a-\frac{37\!\cdots\!15}{18\!\cdots\!96}$ 74587918422210796297/1148020003544252220143826256*a^18 + 2460451160916697275885/2296040007088504440287652512*a^16 + 144837904462356585100901/4592080014177008880575305024*a^14 + 4124588540174671767463257/2296040007088504440287652512*a^12 + 227768103202954312799069555/4592080014177008880575305024*a^10 + 1787101800677070242522393967/2296040007088504440287652512*a^8 + 35011448652712013497253018999/4592080014177008880575305024*a^6 + 107434837618740476831580314791/2296040007088504440287652512*a^4 + 762858217409098783268765778629/4592080014177008880575305024*a^2 + 150132460770490525254780248553/574010001772126110071913128, 369343120542777642205/9184160028354017761150610048*a^18 + 47496146226194744670/71751250221515763758989141*a^16 + 89535649730324922926205/4592080014177008880575305024*a^14 + 2551830794977647044128183/2296040007088504440287652512*a^12 + 35204850565693060324004625/1148020003544252220143826256*a^10 + 1103761900331972389260168525/2296040007088504440287652512*a^8 + 21609272025045552684086040667/4592080014177008880575305024*a^6 + 66359530255138504605530352165/2296040007088504440287652512*a^4 + 945273633341460241398293343819/9184160028354017761150610048*a^2 + 375705332243605526428832941999/2296040007088504440287652512, 93810426043362253401/18368320056708035522301220096*a^18 + 975568902440661830265/9184160028354017761150610048*a^16 + 6751183676454552410855/2296040007088504440287652512*a^14 + 696958626226339029452165/4592080014177008880575305024*a^12 + 41615160441363239422069961/9184160028354017761150610048*a^10 + 91968833767947947925688167/1148020003544252220143826256*a^8 + 511566594695279907395133851/574010001772126110071913128*a^6 + 29091882875321730990388530955/4592080014177008880575305024*a^4 + 491550656905223777163991553197/18368320056708035522301220096*a^2 + 470357798733673502824406974575/9184160028354017761150610048, 93810426043362253401/18368320056708035522301220096*a^18 + 975568902440661830265/9184160028354017761150610048*a^16 + 6751183676454552410855/2296040007088504440287652512*a^14 + 696958626226339029452165/4592080014177008880575305024*a^12 + 41615160441363239422069961/9184160028354017761150610048*a^10 + 91968833767947947925688167/1148020003544252220143826256*a^8 + 511566594695279907395133851/574010001772126110071913128*a^6 + 29091882875321730990388530955/4592080014177008880575305024*a^4 + 491550656905223777163991553197/18368320056708035522301220096*a^2 + 479541958762027520585557584623/9184160028354017761150610048, 12926521540260323352013733/1367273455901147686155774769675904*a^19 + 23912065646596716425/4592080014177008880575305024*a^18 + 97900822095165393521353111/683636727950573843077887384837952*a^17 + 1354540570718333012595/18368320056708035522301220096*a^16 + 3207845628529950190042210149/683636727950573843077887384837952*a^15 + 10455449247133603990157/4592080014177008880575305024*a^14 + 175033126532342121547835550647/683636727950573843077887384837952*a^13 + 635062078739356456256951/4592080014177008880575305024*a^12 + 1196293283159792655964539925853/170909181987643460769471846209488*a^11 + 16668179324812029188835221/4592080014177008880575305024*a^10 + 75418567234812880598557588659303/683636727950573843077887384837952*a^9 + 478086272044065601009511217/9184160028354017761150610048*a^8 + 751025535121835965906557736627427/683636727950573843077887384837952*a^7 + 2089674395582699074833590007/4592080014177008880575305024*a^6 + 4754668352469026639193608354909573/683636727950573843077887384837952*a^5 + 11041436013873567347622851519/4592080014177008880575305024*a^4 + 35804198078755453467711457153681155/1367273455901147686155774769675904*a^3 + 15773648116891872661350643891/2296040007088504440287652512*a^2 + 15593903280735237180102583164706975/341818363975286921538943692418976*a + 166988989695137178013358616323/18368320056708035522301220096, 17980155613901553442740801/341818363975286921538943692418976*a^19 - 322263739335439901613/4592080014177008880575305024*a^18 + 765096905181981787062156581/1367273455901147686155774769675904*a^17 - 21038149858051911219675/18368320056708035522301220096*a^16 + 16409475367441085292494885211/683636727950573843077887384837952*a^15 - 77646676854745094545529/2296040007088504440287652512*a^14 + 904896428783427651465244561135/683636727950573843077887384837952*a^13 - 8884239159088699991183465/4592080014177008880575305024*a^12 + 22661528621784323917404934487301/683636727950573843077887384837952*a^11 - 30554535315970792748488097/574010001772126110071913128*a^10 + 162772082542498316384242429052007/341818363975286921538943692418976*a^9 - 7626493474752346571099087085/9184160028354017761150610048*a^8 + 2926217637344857454385682719664897/683636727950573843077887384837952*a^7 - 18550561524147356286043304503/2296040007088504440287652512*a^6 + 16418278765917296123818409823970969/683636727950573843077887384837952*a^5 - 225911111251354521010783481101/4592080014177008880575305024*a^4 + 52889122674698296855798010851240853/683636727950573843077887384837952*a^3 - 794405513642882528591467066411/4592080014177008880575305024*a^2 + 149702566147308451887350000741234895/1367273455901147686155774769675904*a - 4952859414294125950644025349923/18368320056708035522301220096, 137109966896737649584523063/1367273455901147686155774769675904*a^19 + 1382865383385121859853/18368320056708035522301220096*a^18 + 846503389583145084757713787/683636727950573843077887384837952*a^17 + 22989287662933234880205/18368320056708035522301220096*a^16 + 32180878315754982290301617625/683636727950573843077887384837952*a^15 + 2637433141599988967387/71751250221515763758989141*a^14 + 1773354955959585229775803016843/683636727950573843077887384837952*a^13 + 4790598892657519510317815/2296040007088504440287652512*a^12 + 23006403766110011158753224272467/341818363975286921538943692418976*a^11 + 530487725496895923397879513/9184160028354017761150610048*a^10 + 685031588703141334834150771790623/683636727950573843077887384837952*a^9 + 8362244144895930154504592421/9184160028354017761150610048*a^8 + 6396740511382828384790826960138143/683636727950573843077887384837952*a^7 + 20596827902928475915623839907/2296040007088504440287652512*a^6 + 37493897342592576096007830646626521/683636727950573843077887384837952*a^5 + 31875374265834531500146501507/574010001772126110071913128*a^4 + 254365825258298453244059143776687469/1367273455901147686155774769675904*a^3 + 3669172711476753891529859818841/18368320056708035522301220096*a^2 + 95756603983774084674061682925161125/341818363975286921538943692418976*a + 5930311651874889027337441739265/18368320056708035522301220096, 125987668018153603103985573/2734546911802295372311549539351808*a^19 - 2025903361884290278563/18368320056708035522301220096*a^18 + 1722978694058135894290720751/2734546911802295372311549539351808*a^17 - 16896880263060378251565/9184160028354017761150610048*a^16 + 1894029490664122054765601971/85454590993821730384735923104744*a^15 - 15492245096599413303051/287005000886063055035956564*a^14 + 104058662953552289390415620797/85454590993821730384735923104744*a^13 - 14049797296530976652635045/4592080014177008880575305024*a^12 + 44320032889269783249758334355161/1367273455901147686155774769675904*a^11 - 778790171372816347612246071/9184160028354017761150610048*a^10 + 677765829389265349803953095333243/1367273455901147686155774769675904*a^9 - 1537400684272469263816969413/1148020003544252220143826256*a^8 + 1629432344974790613368126300864779/341818363975286921538943692418976*a^7 - 30356626717659902720250065237/2296040007088504440287652512*a^6 + 9880199078158258347729465572116633/341818363975286921538943692418976*a^5 - 376680618623079693864609864867/4592080014177008880575305024*a^4 + 279490977672090634281254748061544545/2734546911802295372311549539351808*a^3 - 5433530793224539393035641355351/18368320056708035522301220096*a^2 + 443758783181432996174724524186466435/2734546911802295372311549539351808*a - 4384482660064298030377373329467/9184160028354017761150610048, 100134624937632956399958107/2734546911802295372311549539351808*a^19 - 1003948006125623621651/18368320056708035522301220096*a^18 + 1331375405677474320205308307/2734546911802295372311549539351808*a^17 - 16377930911733921533955/18368320056708035522301220096*a^16 + 11944390296783026248082605619/683636727950573843077887384837952*a^15 - 60440043931156938144517/2296040007088504440287652512*a^14 + 657436177096076193575489415729/683636727950573843077887384837952*a^13 - 3458578187691824024608699/2296040007088504440287652512*a^12 + 34749686623991442002042014948337/1367273455901147686155774769675904*a^11 - 380584687314921327798398707/9184160028354017761150610048*a^10 + 526928694919639588606837918014637/1367273455901147686155774769675904*a^9 - 5934570260520631785674559315/9184160028354017761150610048*a^8 + 2507839154827745260829694865102131/683636727950573843077887384837952*a^7 - 3603655187445884395407294773/574010001772126110071913128*a^6 + 15005729803847490056265322789323693/683636727950573843077887384837952*a^5 - 43684089087071413831287311777/1148020003544252220143826256*a^4 + 207882581514579727345831833754182235/2734546911802295372311549539351808*a^3 - 2433693027796036374620266410191/18368320056708035522301220096*a^2 + 319007556935551098733903858868810635/2734546911802295372311549539351808*a - 3733270797606628695535397828415/18368320056708035522301220096 (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:		\( 1589230.0087159988 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1589230.0087159988 \cdot 4792880}{2\cdot\sqrt{3525175555633378160676822382018560000000000}}\cr\approx \mathstrut & 0.194518819666524 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_{10}$ (as 20T3):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

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		Deg $20$	$4$	$5$	$40$
\(3\) 3	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	3.10.5.2	$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
\(5\) 5		Deg $20$	$2$	$10$	$10$
\(11\) 11	11.10.9.7	$x^{10} + 11$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
	11.10.9.7	$x^{10} + 11$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$