↑

Properties

Label 27.1.786...199.1
Degree $27$
Signature $[1, 13]$
Discriminant $-7.864\times 10^{40}$
Root discriminant \(32.71\)
Ramified prime $1399$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $D_{27}$ (as 27T8)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1)
 
gp: K = bnfinit(y^27 - y^26 + 6*y^25 - 31*y^24 + 97*y^23 - 269*y^22 + 599*y^21 - 994*y^20 + 1307*y^19 - 1298*y^18 + 592*y^17 + 817*y^16 - 2461*y^15 + 4001*y^14 - 4903*y^13 + 4315*y^12 - 2279*y^11 - 512*y^10 + 3872*y^9 - 6814*y^8 + 7826*y^7 - 7115*y^6 + 5689*y^5 - 3842*y^4 + 1951*y^3 - 627*y^2 + 101*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1)
 

\( x^{27} - x^{26} + 6 x^{25} - 31 x^{24} + 97 x^{23} - 269 x^{22} + 599 x^{21} - 994 x^{20} + 1307 x^{19} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-78637606867438430727852801672920631138199\) \(\medspace = -\,1399^{13}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(32.71\)
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:  $1399^{1/2}\approx 37.403208418530085$
Ramified primes:   \(1399\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1399}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{12}-\frac{2}{9}a^{4}$, $\frac{1}{81}a^{21}+\frac{1}{27}a^{20}+\frac{1}{27}a^{19}-\frac{1}{81}a^{18}+\frac{1}{27}a^{17}+\frac{4}{81}a^{15}-\frac{1}{27}a^{14}+\frac{1}{9}a^{13}+\frac{11}{81}a^{12}+\frac{1}{27}a^{11}+\frac{4}{27}a^{10}+\frac{1}{81}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{29}{81}a^{6}-\frac{4}{9}a^{5}-\frac{5}{27}a^{4}-\frac{5}{81}a^{3}-\frac{4}{27}a^{2}+\frac{8}{27}a+\frac{23}{81}$, $\frac{1}{1053}a^{22}-\frac{2}{351}a^{21}+\frac{16}{351}a^{20}-\frac{37}{1053}a^{19}+\frac{4}{351}a^{18}+\frac{1}{39}a^{17}-\frac{32}{1053}a^{16}-\frac{4}{351}a^{15}-\frac{4}{117}a^{14}-\frac{88}{1053}a^{13}+\frac{19}{351}a^{12}+\frac{25}{351}a^{11}-\frac{89}{1053}a^{10}-\frac{38}{351}a^{9}-\frac{28}{351}a^{8}-\frac{43}{1053}a^{7}-\frac{35}{117}a^{6}-\frac{158}{351}a^{5}+\frac{301}{1053}a^{4}-\frac{175}{351}a^{3}-\frac{19}{351}a^{2}-\frac{373}{1053}a-\frac{50}{117}$, $\frac{1}{1053}a^{23}-\frac{1}{1053}a^{21}-\frac{22}{1053}a^{20}-\frac{5}{351}a^{19}-\frac{5}{1053}a^{18}-\frac{2}{81}a^{17}+\frac{10}{351}a^{16}-\frac{43}{1053}a^{15}-\frac{31}{1053}a^{14}-\frac{40}{351}a^{13}-\frac{77}{1053}a^{12}-\frac{29}{1053}a^{11}+\frac{5}{351}a^{10}-\frac{79}{1053}a^{9}-\frac{118}{1053}a^{8}+\frac{17}{351}a^{7}-\frac{518}{1053}a^{6}+\frac{265}{1053}a^{5}+\frac{34}{117}a^{4}-\frac{217}{1053}a^{3}+\frac{11}{81}a^{2}+\frac{92}{351}a-\frac{191}{1053}$, $\frac{1}{1053}a^{24}-\frac{2}{1053}a^{21}-\frac{2}{351}a^{20}+\frac{4}{117}a^{19}-\frac{40}{1053}a^{18}+\frac{2}{117}a^{17}+\frac{14}{351}a^{16}-\frac{56}{1053}a^{15}+\frac{23}{351}a^{13}-\frac{37}{1053}a^{12}-\frac{22}{351}a^{11}+\frac{16}{117}a^{10}-\frac{89}{1053}a^{9}+\frac{5}{117}a^{8}-\frac{2}{39}a^{7}+\frac{2}{1053}a^{6}+\frac{139}{351}a^{5}-\frac{34}{117}a^{4}+\frac{190}{1053}a^{3}+\frac{164}{351}a^{2}-\frac{175}{351}a-\frac{437}{1053}$, $\frac{1}{9499113}a^{25}+\frac{2179}{9499113}a^{24}-\frac{770}{3166371}a^{23}-\frac{142}{351819}a^{22}+\frac{22108}{9499113}a^{21}+\frac{5609}{3166371}a^{20}+\frac{163921}{3166371}a^{19}+\frac{7163}{730701}a^{18}+\frac{54868}{1055457}a^{17}-\frac{164327}{3166371}a^{16}+\frac{13225}{306423}a^{15}+\frac{1615}{102141}a^{14}-\frac{142777}{1055457}a^{13}-\frac{1384609}{9499113}a^{12}+\frac{43298}{3166371}a^{11}+\frac{367999}{3166371}a^{10}+\frac{1261300}{9499113}a^{9}+\frac{26657}{1055457}a^{8}+\frac{258833}{3166371}a^{7}-\frac{2493631}{9499113}a^{6}+\frac{1159861}{3166371}a^{5}+\frac{456713}{1055457}a^{4}+\frac{4384573}{9499113}a^{3}+\frac{111658}{3166371}a^{2}+\frac{701534}{9499113}a-\frac{589700}{9499113}$, $\frac{1}{3787305852213}a^{26}-\frac{149548}{3787305852213}a^{25}+\frac{791939755}{3787305852213}a^{24}+\frac{160117921}{420811761357}a^{23}+\frac{67194607}{3787305852213}a^{22}+\frac{561529180}{291331219401}a^{21}+\frac{4954417159}{97110406467}a^{20}+\frac{172818018944}{3787305852213}a^{19}-\frac{126182037067}{3787305852213}a^{18}+\frac{17211615944}{420811761357}a^{17}-\frac{6233087738}{291331219401}a^{16}+\frac{5425039036}{122171156523}a^{15}-\frac{5801079697}{1262435284071}a^{14}-\frac{236014530691}{3787305852213}a^{13}+\frac{11339074948}{80580975579}a^{12}-\frac{12584457790}{1262435284071}a^{11}-\frac{629115830345}{3787305852213}a^{10}+\frac{331516908106}{3787305852213}a^{9}+\frac{41728899608}{420811761357}a^{8}-\frac{318454670992}{3787305852213}a^{7}+\frac{258088011470}{3787305852213}a^{6}+\frac{63213724585}{140270587119}a^{5}+\frac{829334261707}{3787305852213}a^{4}+\frac{638339062513}{3787305852213}a^{3}+\frac{1149502756733}{3787305852213}a^{2}+\frac{553883555774}{1262435284071}a-\frac{1471759457537}{3787305852213}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{723803975812}{3787305852213}a^{26}-\frac{524707497172}{3787305852213}a^{25}+\frac{4047237270289}{3787305852213}a^{24}-\frac{7105472913107}{1262435284071}a^{23}+\frac{63498621481159}{3787305852213}a^{22}-\frac{173336080013948}{3787305852213}a^{21}+\frac{41692112627959}{420811761357}a^{20}-\frac{586853826215887}{3787305852213}a^{19}+\frac{725392138007672}{3787305852213}a^{18}-\frac{218448907035557}{1262435284071}a^{17}+\frac{150073810684066}{3787305852213}a^{16}+\frac{22835038826164}{122171156523}a^{15}-\frac{523852124791829}{1262435284071}a^{14}+\frac{23\!\cdots\!64}{3787305852213}a^{13}-\frac{56747623802339}{80580975579}a^{12}+\frac{7368533826725}{13574572947}a^{11}-\frac{738564145039865}{3787305852213}a^{10}-\frac{786337946801909}{3787305852213}a^{9}+\frac{871149039484331}{1262435284071}a^{8}-\frac{40\!\cdots\!59}{3787305852213}a^{7}+\frac{317246093495909}{291331219401}a^{6}-\frac{88489303396433}{97110406467}a^{5}+\frac{26\!\cdots\!73}{3787305852213}a^{4}-\frac{16\!\cdots\!84}{3787305852213}a^{3}+\frac{661495941808160}{3787305852213}a^{2}-\frac{38651281898165}{1262435284071}a-\frac{247665876353}{291331219401}$, $\frac{66758092084}{3787305852213}a^{26}-\frac{4627708513}{3787305852213}a^{25}+\frac{353969531608}{3787305852213}a^{24}-\frac{560699184250}{1262435284071}a^{23}+\frac{4692067671727}{3787305852213}a^{22}-\frac{12166814900411}{3787305852213}a^{21}+\frac{8133489155339}{1262435284071}a^{20}-\frac{2455363626214}{291331219401}a^{19}+\frac{31293799689515}{3787305852213}a^{18}-\frac{5055121228414}{1262435284071}a^{17}-\frac{2128387635707}{291331219401}a^{16}+\frac{2544735809374}{122171156523}a^{15}-\frac{11760179957900}{420811761357}a^{14}+\frac{119971328687810}{3787305852213}a^{13}-\frac{167543770241}{6198536583}a^{12}+\frac{8081460645748}{1262435284071}a^{11}+\frac{65555457770605}{3787305852213}a^{10}-\frac{124778915190383}{3787305852213}a^{9}+\frac{65664266299570}{1262435284071}a^{8}-\frac{217843294853188}{3787305852213}a^{7}+\frac{136716551207588}{3787305852213}a^{6}-\frac{1473752014318}{97110406467}a^{5}+\frac{25374498846079}{3787305852213}a^{4}+\frac{1613452918246}{291331219401}a^{3}-\frac{45653818659343}{3787305852213}a^{2}+\frac{12301020231856}{1262435284071}a-\frac{10510242423770}{3787305852213}$, $\frac{1480149941}{222782697189}a^{26}+\frac{8119564177}{222782697189}a^{25}+\frac{13653382403}{222782697189}a^{24}+\frac{1722430393}{24753633021}a^{23}-\frac{63979072489}{222782697189}a^{22}+\frac{200472685343}{222782697189}a^{21}-\frac{2501783095}{765576279}a^{20}+\frac{1652990130124}{222782697189}a^{19}-\frac{2375963852765}{222782697189}a^{18}+\frac{341829197284}{24753633021}a^{17}-\frac{2577182205337}{222782697189}a^{16}-\frac{282329833}{552810663}a^{15}+\frac{1221989682832}{74260899063}a^{14}-\frac{6620909311682}{222782697189}a^{13}+\frac{213642658700}{4740057387}a^{12}-\frac{3520420375310}{74260899063}a^{11}+\frac{6411244530488}{222782697189}a^{10}-\frac{1323041603539}{222782697189}a^{9}-\frac{15485682883}{798504291}a^{8}+\frac{13069956870127}{222782697189}a^{7}-\frac{16716664293662}{222782697189}a^{6}+\frac{1595673343744}{24753633021}a^{5}-\frac{11744487277582}{222782697189}a^{4}+\frac{9086851865762}{222782697189}a^{3}-\frac{3892868277620}{222782697189}a^{2}+\frac{23947255700}{8251211007}a+\frac{111929764604}{222782697189}$, $\frac{27400364093}{3787305852213}a^{26}+\frac{401256435514}{3787305852213}a^{25}+\frac{518627434787}{3787305852213}a^{24}+\frac{223729334521}{420811761357}a^{23}-\frac{5732489618647}{3787305852213}a^{22}+\frac{16473967170887}{3787305852213}a^{21}-\frac{1247046340837}{97110406467}a^{20}+\frac{91598022696541}{3787305852213}a^{19}-\frac{121271297908445}{3787305852213}a^{18}+\frac{16050410006452}{420811761357}a^{17}-\frac{90636813187456}{3787305852213}a^{16}-\frac{1805880866971}{122171156523}a^{15}+\frac{71504801561761}{1262435284071}a^{14}-\frac{358104537260993}{3787305852213}a^{13}+\frac{10624703649122}{80580975579}a^{12}-\frac{153574777195928}{1262435284071}a^{11}+\frac{240485292867149}{3787305852213}a^{10}-\frac{6591799694656}{3787305852213}a^{9}-\frac{34099802215676}{420811761357}a^{8}+\frac{688885830336706}{3787305852213}a^{7}-\frac{59035484503727}{291331219401}a^{6}+\frac{75086987439379}{420811761357}a^{5}-\frac{560244988558396}{3787305852213}a^{4}+\frac{379223882722946}{3787305852213}a^{3}-\frac{5946040185224}{122171156523}a^{2}+\frac{12944809290995}{1262435284071}a-\frac{3792647459425}{3787305852213}$, $\frac{103669695616}{1262435284071}a^{26}-\frac{8533754888}{140270587119}a^{25}+\frac{591747866279}{1262435284071}a^{24}-\frac{3048295959776}{1262435284071}a^{23}+\frac{9218208011765}{1262435284071}a^{22}-\frac{25166632145380}{1262435284071}a^{21}+\frac{54603822671675}{1262435284071}a^{20}-\frac{86353677629948}{1262435284071}a^{19}+\frac{107534195733280}{1262435284071}a^{18}-\frac{98384740718915}{1262435284071}a^{17}+\frac{26556116885417}{1262435284071}a^{16}+\frac{3196920035204}{40723718841}a^{15}-\frac{228806153586064}{1262435284071}a^{14}+\frac{342024523289707}{1262435284071}a^{13}-\frac{8432931158593}{26860325193}a^{12}+\frac{313793618512762}{1262435284071}a^{11}-\frac{121433103808261}{1262435284071}a^{10}-\frac{106214965636438}{1262435284071}a^{9}+\frac{374929677855698}{1262435284071}a^{8}-\frac{593164748953262}{1262435284071}a^{7}+\frac{618748393644265}{1262435284071}a^{6}-\frac{16824086074043}{40723718841}a^{5}+\frac{406355601480935}{1262435284071}a^{4}-\frac{256920346807126}{1262435284071}a^{3}+\frac{105932645413954}{1262435284071}a^{2}-\frac{21438359163041}{1262435284071}a-\frac{309861542374}{1262435284071}$, $\frac{292952105414}{3787305852213}a^{26}-\frac{103960177871}{1262435284071}a^{25}+\frac{1636409708548}{3787305852213}a^{24}-\frac{3074880320971}{1262435284071}a^{23}+\frac{28162643884517}{3787305852213}a^{22}-\frac{77299728982745}{3787305852213}a^{21}+\frac{704714978751}{15585620791}a^{20}-\frac{21303605128196}{291331219401}a^{19}+\frac{351135895218272}{3787305852213}a^{18}-\frac{37060891400003}{420811761357}a^{17}+\frac{114728791879577}{3787305852213}a^{16}+\frac{9345602445802}{122171156523}a^{15}-\frac{80108485543195}{420811761357}a^{14}+\frac{11\!\cdots\!23}{3787305852213}a^{13}-\frac{27811763439956}{80580975579}a^{12}+\frac{357478724391713}{1262435284071}a^{11}-\frac{460032343491388}{3787305852213}a^{10}-\frac{282721829446592}{3787305852213}a^{9}+\frac{129507908038357}{420811761357}a^{8}-\frac{61772127345623}{122171156523}a^{7}+\frac{20\!\cdots\!08}{3787305852213}a^{6}-\frac{45085829820403}{97110406467}a^{5}+\frac{13\!\cdots\!62}{3787305852213}a^{4}-\frac{888037742740256}{3787305852213}a^{3}+\frac{392206154384293}{3787305852213}a^{2}-\frac{90205326658544}{3787305852213}a+\frac{5844224856232}{3787305852213}$, $\frac{23678558528}{222782697189}a^{26}-\frac{20466439628}{222782697189}a^{25}+\frac{4346963831}{7186538619}a^{24}-\frac{238395449714}{74260899063}a^{23}+\frac{167340558296}{17137130553}a^{22}-\frac{5954126258770}{222782697189}a^{21}+\frac{4351638857743}{74260899063}a^{20}-\frac{20879002482833}{222782697189}a^{19}+\frac{26319788513182}{222782697189}a^{18}-\frac{8208934039814}{74260899063}a^{17}+\frac{7742882320643}{222782697189}a^{16}+\frac{730840339130}{7186538619}a^{15}-\frac{6076748691329}{24753633021}a^{14}+\frac{83199754326979}{222782697189}a^{13}-\frac{2075684470972}{4740057387}a^{12}+\frac{26285681443610}{74260899063}a^{11}-\frac{32755310890996}{222782697189}a^{10}-\frac{23004707038471}{222782697189}a^{9}+\frac{29827073365007}{74260899063}a^{8}-\frac{11085309778349}{17137130553}a^{7}+\frac{152870371839697}{222782697189}a^{6}-\frac{43538812393910}{74260899063}a^{5}+\frac{101731816014278}{222782697189}a^{4}-\frac{64554746679232}{222782697189}a^{3}+\frac{28062296278435}{222782697189}a^{2}-\frac{227487051112}{8251211007}a+\frac{218488751699}{222782697189}$, $\frac{91878899}{7186538619}a^{26}-\frac{7732124225}{222782697189}a^{25}+\frac{14949785189}{222782697189}a^{24}-\frac{12937536161}{24753633021}a^{23}+\frac{386869355996}{222782697189}a^{22}-\frac{1056801710947}{222782697189}a^{21}+\frac{831258024572}{74260899063}a^{20}-\frac{4317697462733}{222782697189}a^{19}+\frac{5613016635079}{222782697189}a^{18}-\frac{631391303518}{24753633021}a^{17}+\frac{2762109066689}{222782697189}a^{16}+\frac{120551150258}{7186538619}a^{15}-\frac{118892495666}{2395512873}a^{14}+\frac{17180023588150}{222782697189}a^{13}-\frac{452485685614}{4740057387}a^{12}+\frac{6302307323281}{74260899063}a^{11}-\frac{8816130493639}{222782697189}a^{10}-\frac{3405152185072}{222782697189}a^{9}+\frac{1850831771213}{24753633021}a^{8}-\frac{30613334611235}{222782697189}a^{7}+\frac{34659537703327}{222782697189}a^{6}-\frac{3187272343958}{24753633021}a^{5}+\frac{22714411223288}{222782697189}a^{4}-\frac{15539681980048}{222782697189}a^{3}+\frac{6509508002539}{222782697189}a^{2}-\frac{449871959704}{74260899063}a-\frac{190090741501}{222782697189}$, $\frac{6513541013}{80580975579}a^{26}-\frac{1346412166}{80580975579}a^{25}+\frac{11734988666}{26860325193}a^{24}-\frac{1487422231}{688726287}a^{23}+\frac{478410122651}{80580975579}a^{22}-\frac{433795603733}{26860325193}a^{21}+\frac{892553102810}{26860325193}a^{20}-\frac{3829338675104}{80580975579}a^{19}+\frac{492762732364}{8953441731}a^{18}-\frac{1169763322252}{26860325193}a^{17}-\frac{514770849295}{80580975579}a^{16}+\frac{64782105608}{866462103}a^{15}-\frac{1189144791988}{8953441731}a^{14}+\frac{14969742814219}{80580975579}a^{13}-\frac{5288342282186}{26860325193}a^{12}+\frac{3307557951350}{26860325193}a^{11}-\frac{1397609406463}{80580975579}a^{10}-\frac{274254092171}{2984480577}a^{9}+\frac{6355051417378}{26860325193}a^{8}-\frac{832098129977}{2599386309}a^{7}+\frac{7611606876032}{26860325193}a^{6}-\frac{2071124513710}{8953441731}a^{5}+\frac{1108268449052}{6198536583}a^{4}-\frac{190987711330}{2066178861}a^{3}+\frac{2193249739801}{80580975579}a^{2}-\frac{259531582717}{80580975579}a+\frac{9530919871}{8953441731}$, $\frac{225875507413}{3787305852213}a^{26}-\frac{368094970045}{1262435284071}a^{25}+\frac{1414287320441}{3787305852213}a^{24}-\frac{3976288572337}{1262435284071}a^{23}+\frac{44518731904627}{3787305852213}a^{22}-\frac{123250684664839}{3787305852213}a^{21}+\frac{7841852979437}{97110406467}a^{20}-\frac{570854640816388}{3787305852213}a^{19}+\frac{788610578123530}{3787305852213}a^{18}-\frac{288448885100612}{1262435284071}a^{17}+\frac{18511164491944}{122171156523}a^{16}+\frac{8712983628317}{122171156523}a^{15}-\frac{449250906937519}{1262435284071}a^{14}+\frac{23\!\cdots\!85}{3787305852213}a^{13}-\frac{65006711397442}{80580975579}a^{12}+\frac{998597482157837}{1262435284071}a^{11}-\frac{17\!\cdots\!19}{3787305852213}a^{10}+\frac{29989793941892}{3787305852213}a^{9}+\frac{634964709104324}{1262435284071}a^{8}-\frac{40\!\cdots\!33}{3787305852213}a^{7}+\frac{50\!\cdots\!99}{3787305852213}a^{6}-\frac{15\!\cdots\!71}{1262435284071}a^{5}+\frac{36\!\cdots\!80}{3787305852213}a^{4}-\frac{26\!\cdots\!04}{3787305852213}a^{3}+\frac{13\!\cdots\!37}{3787305852213}a^{2}-\frac{393427615698913}{3787305852213}a+\frac{359437236992}{122171156523}$, $\frac{984363413}{40723718841}a^{26}+\frac{138253167512}{3787305852213}a^{25}+\frac{554202524318}{3787305852213}a^{24}-\frac{508780378810}{1262435284071}a^{23}+\frac{25505490451}{32370135489}a^{22}-\frac{8891960068924}{3787305852213}a^{21}+\frac{3381188084305}{1262435284071}a^{20}-\frac{690638704501}{1262435284071}a^{19}-\frac{9743964651650}{3787305852213}a^{18}+\frac{4090309746814}{420811761357}a^{17}-\frac{23970491400925}{1262435284071}a^{16}+\frac{2217607772987}{122171156523}a^{15}-\frac{482203054054}{40723718841}a^{14}+\frac{275272352129}{140270587119}a^{13}+\frac{1463340741362}{80580975579}a^{12}-\frac{52564151950736}{1262435284071}a^{11}+\frac{54838256185835}{1262435284071}a^{10}-\frac{161180673547318}{3787305852213}a^{9}+\frac{14027763759052}{420811761357}a^{8}+\frac{2620183252501}{1262435284071}a^{7}-\frac{142488730442192}{3787305852213}a^{6}+\frac{58623624081005}{1262435284071}a^{5}-\frac{22992586702552}{420811761357}a^{4}+\frac{170524129499270}{3787305852213}a^{3}-\frac{15384616678828}{420811761357}a^{2}+\frac{60164789116075}{3787305852213}a-\frac{22022420899267}{3787305852213}$, $\frac{273499476239}{3787305852213}a^{26}-\frac{15772031879}{1262435284071}a^{25}+\frac{1782410133388}{3787305852213}a^{24}-\frac{771632766485}{420811761357}a^{23}+\frac{21847512554378}{3787305852213}a^{22}-\frac{59746035509006}{3787305852213}a^{21}+\frac{3221903272133}{97110406467}a^{20}-\frac{199368106676015}{3787305852213}a^{19}+\frac{252962200943486}{3787305852213}a^{18}-\frac{76188392250455}{1262435284071}a^{17}+\frac{68260210315496}{3787305852213}a^{16}+\frac{6863630171407}{122171156523}a^{15}-\frac{174801392006554}{1262435284071}a^{14}+\frac{798791798029492}{3787305852213}a^{13}-\frac{19566776276315}{80580975579}a^{12}+\frac{83375135943407}{420811761357}a^{11}-\frac{331045498100716}{3787305852213}a^{10}-\frac{226252040514047}{3787305852213}a^{9}+\frac{284821806691061}{1262435284071}a^{8}-\frac{13\!\cdots\!18}{3787305852213}a^{7}+\frac{14\!\cdots\!68}{3787305852213}a^{6}-\frac{15688239394730}{46756862373}a^{5}+\frac{982270095313163}{3787305852213}a^{4}-\frac{636993127926602}{3787305852213}a^{3}+\frac{284276032125166}{3787305852213}a^{2}-\frac{78137692429358}{3787305852213}a+\frac{83211457375}{291331219401}$, $\frac{295849580054}{3787305852213}a^{26}-\frac{132315754105}{3787305852213}a^{25}+\frac{43447370098}{97110406467}a^{24}-\frac{2735565298546}{1262435284071}a^{23}+\frac{24170430138503}{3787305852213}a^{22}-\frac{21938850300205}{1262435284071}a^{21}+\frac{1727814010375}{46756862373}a^{20}-\frac{214257320348756}{3787305852213}a^{19}+\frac{87520345351547}{1262435284071}a^{18}-\frac{77006985009446}{1262435284071}a^{17}+\frac{39408291363035}{3787305852213}a^{16}+\frac{968831377351}{13574572947}a^{15}-\frac{191889552672536}{1262435284071}a^{14}+\frac{850078940896921}{3787305852213}a^{13}-\frac{83982754353}{331608953}a^{12}+\frac{18448989953500}{97110406467}a^{11}-\frac{246949523806045}{3787305852213}a^{10}-\frac{100504688241401}{1262435284071}a^{9}+\frac{324536825490668}{1262435284071}a^{8}-\frac{14\!\cdots\!28}{3787305852213}a^{7}+\frac{490175483088754}{1262435284071}a^{6}-\frac{412918416444472}{1262435284071}a^{5}+\frac{961531291037648}{3787305852213}a^{4}-\frac{2072564214701}{13574572947}a^{3}+\frac{232560822084277}{3787305852213}a^{2}-\frac{44493778335319}{3787305852213}a+\frac{100910172938}{420811761357}$ Copy content Toggle raw display (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:  \( 6401978481.787342 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 6401978481.787342 \cdot 2}{2\cdot\sqrt{78637606867438430727852801672920631138199}}\cr\approx \mathstrut & 1.08609383919128 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{27}$ (as 27T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.1399.1, 9.1.3830635754401.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $27$ ${\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} }$ $27$ ${\href{/padicField/7.9.0.1}{9} }^{3}$ $27$ ${\href{/padicField/13.2.0.1}{2} }^{13}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $27$ ${\href{/padicField/23.9.0.1}{9} }^{3}$ $27$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{9}$ $27$ ${\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.3.0.1}{3} }^{9}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(1399\) Copy content Toggle raw display $\Q_{1399}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1399.2t1.a.a$1$ $ 1399 $ \(\Q(\sqrt{-1399}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1399.3t2.a.a$2$ $ 1399 $ 3.1.1399.1 $S_3$ (as 3T2) $1$ $0$
* 2.1399.9t3.a.b$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.9t3.a.c$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.9t3.a.a$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.27t8.a.g$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.i$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.d$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.f$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.h$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.b$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.c$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.e$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.a$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.