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Properties

Label 27.1.108...375.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.090\times 10^{48}$
Root discriminant \(60.14\)
Ramified primes $5,7,67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625)
 
gp: K = bnfinit(y^27 - 8*y^26 + 11*y^25 + 79*y^24 - 318*y^23 + 272*y^22 + 1817*y^21 - 11487*y^20 + 27792*y^19 - 5214*y^18 - 88324*y^17 + 101476*y^16 + 384139*y^15 - 2894444*y^14 + 8285833*y^13 - 7065937*y^12 - 6208737*y^11 - 6594753*y^10 + 51530498*y^9 - 40765921*y^8 + 72314518*y^7 - 318855243*y^6 + 340132420*y^5 + 140246505*y^4 - 148040525*y^3 - 594053575*y^2 + 805097125*y - 295863625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625)
 

\( x^{27} - 8 x^{26} + 11 x^{25} + 79 x^{24} - 318 x^{23} + 272 x^{22} + 1817 x^{21} - 11487 x^{20} + 27792 x^{19} - 5214 x^{18} - 88324 x^{17} + 101476 x^{16} + 384139 x^{15} + \cdots - 295863625 \) 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:   \(-1089801024238603052304820653163822019730224609375\) \(\medspace = -\,5^{13}\cdot 7^{18}\cdot 67^{13}\) 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:  \(60.14\)
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))
 
Ramified primes:   \(5\), \(7\), \(67\) 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:  \(\Q(\sqrt{-335}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{9}+\frac{1}{5}a^{5}$, $\frac{1}{5}a^{14}+\frac{2}{5}a^{6}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{7}+\frac{2}{5}a^{3}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{8}+\frac{2}{5}a^{4}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{9}+\frac{2}{5}a^{5}$, $\frac{1}{25}a^{18}+\frac{1}{25}a^{17}+\frac{2}{25}a^{16}-\frac{2}{25}a^{15}+\frac{1}{25}a^{14}-\frac{1}{25}a^{13}-\frac{1}{25}a^{11}+\frac{9}{25}a^{9}-\frac{11}{25}a^{8}-\frac{12}{25}a^{7}-\frac{7}{25}a^{6}+\frac{11}{25}a^{5}-\frac{6}{25}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{17}+\frac{1}{25}a^{16}-\frac{2}{25}a^{15}-\frac{2}{25}a^{14}+\frac{1}{25}a^{13}-\frac{1}{25}a^{12}+\frac{1}{25}a^{11}-\frac{1}{25}a^{10}+\frac{1}{5}a^{9}+\frac{9}{25}a^{8}-\frac{1}{5}a^{7}-\frac{12}{25}a^{6}+\frac{8}{25}a^{5}+\frac{11}{25}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{20}+\frac{1}{25}a^{16}+\frac{1}{25}a^{12}-\frac{9}{25}a^{8}+\frac{6}{25}a^{4}$, $\frac{1}{25}a^{21}+\frac{1}{25}a^{17}+\frac{1}{25}a^{13}-\frac{9}{25}a^{9}+\frac{6}{25}a^{5}$, $\frac{1}{2125}a^{22}-\frac{36}{2125}a^{21}-\frac{36}{2125}a^{20}+\frac{3}{425}a^{19}+\frac{43}{2125}a^{17}-\frac{108}{2125}a^{16}-\frac{138}{2125}a^{15}-\frac{2}{25}a^{14}-\frac{41}{425}a^{13}-\frac{46}{2125}a^{12}-\frac{94}{2125}a^{11}-\frac{149}{2125}a^{10}-\frac{137}{425}a^{9}+\frac{18}{425}a^{8}-\frac{863}{2125}a^{7}+\frac{1028}{2125}a^{6}-\frac{287}{2125}a^{5}+\frac{103}{425}a^{4}+\frac{22}{85}a^{3}-\frac{3}{17}a^{2}-\frac{2}{85}a+\frac{1}{17}$, $\frac{1}{1566125}a^{23}+\frac{1}{92125}a^{22}+\frac{27211}{1566125}a^{21}+\frac{28112}{1566125}a^{20}-\frac{52}{28475}a^{19}-\frac{10157}{1566125}a^{18}+\frac{52321}{1566125}a^{17}-\frac{75987}{1566125}a^{16}-\frac{54149}{1566125}a^{15}-\frac{31321}{313225}a^{14}+\frac{129764}{1566125}a^{13}+\frac{71503}{1566125}a^{12}-\frac{131611}{1566125}a^{11}-\frac{107352}{1566125}a^{10}-\frac{14587}{313225}a^{9}-\frac{494108}{1566125}a^{8}+\frac{610639}{1566125}a^{7}-\frac{669068}{1566125}a^{6}-\frac{269781}{1566125}a^{5}+\frac{154744}{313225}a^{4}+\frac{26787}{62645}a^{3}-\frac{17763}{62645}a^{2}+\frac{67}{935}a+\frac{49}{187}$, $\frac{1}{1566125}a^{24}-\frac{347}{1566125}a^{22}-\frac{16596}{1566125}a^{21}-\frac{48}{313225}a^{20}+\frac{5298}{1566125}a^{19}-\frac{5118}{313225}a^{18}-\frac{70726}{1566125}a^{17}+\frac{110757}{1566125}a^{16}-\frac{1976}{28475}a^{15}-\frac{152266}{1566125}a^{14}+\frac{14566}{313225}a^{13}-\frac{30143}{1566125}a^{12}-\frac{5054}{1566125}a^{11}+\frac{22887}{313225}a^{10}+\frac{5102}{1566125}a^{9}+\frac{120998}{313225}a^{8}-\frac{171074}{1566125}a^{7}-\frac{264587}{1566125}a^{6}+\frac{26288}{62645}a^{5}-\frac{18182}{313225}a^{4}+\frac{16963}{62645}a^{3}+\frac{1261}{5695}a^{2}-\frac{16}{187}a+\frac{69}{187}$, $\frac{1}{4019358014375}a^{25}-\frac{1092866}{4019358014375}a^{24}+\frac{476863}{4019358014375}a^{23}+\frac{733574131}{4019358014375}a^{22}+\frac{50923054101}{4019358014375}a^{21}-\frac{70866659987}{4019358014375}a^{20}-\frac{62376008723}{4019358014375}a^{19}+\frac{79492017619}{4019358014375}a^{18}-\frac{146356384442}{4019358014375}a^{17}+\frac{299501636793}{4019358014375}a^{16}-\frac{246968380411}{4019358014375}a^{15}+\frac{220948660616}{4019358014375}a^{14}-\frac{18324496293}{365396183125}a^{13}+\frac{51219224684}{4019358014375}a^{12}+\frac{281579707479}{4019358014375}a^{11}+\frac{16433562326}{236432824375}a^{10}-\frac{54779437001}{236432824375}a^{9}+\frac{114291735318}{236432824375}a^{8}-\frac{2289056378}{4019358014375}a^{7}+\frac{1574165690332}{4019358014375}a^{6}-\frac{71883675883}{160774320575}a^{5}+\frac{205226591736}{803871602875}a^{4}+\frac{2050424108}{14615847325}a^{3}+\frac{737604766}{3420730225}a^{2}+\frac{210878394}{479923345}a+\frac{112787458}{479923345}$, $\frac{1}{45\!\cdots\!75}a^{26}-\frac{45\!\cdots\!16}{45\!\cdots\!75}a^{25}-\frac{41\!\cdots\!02}{45\!\cdots\!75}a^{24}-\frac{13\!\cdots\!64}{45\!\cdots\!75}a^{23}-\frac{18\!\cdots\!49}{45\!\cdots\!75}a^{22}-\frac{17\!\cdots\!24}{19\!\cdots\!25}a^{21}-\frac{19\!\cdots\!51}{19\!\cdots\!25}a^{20}+\frac{28\!\cdots\!49}{45\!\cdots\!75}a^{19}+\frac{87\!\cdots\!48}{45\!\cdots\!75}a^{18}-\frac{19\!\cdots\!32}{45\!\cdots\!75}a^{17}+\frac{26\!\cdots\!69}{45\!\cdots\!75}a^{16}+\frac{13\!\cdots\!43}{24\!\cdots\!25}a^{15}+\frac{65\!\cdots\!17}{45\!\cdots\!75}a^{14}+\frac{18\!\cdots\!54}{45\!\cdots\!75}a^{13}+\frac{23\!\cdots\!62}{67\!\cdots\!25}a^{12}+\frac{57\!\cdots\!07}{45\!\cdots\!75}a^{11}-\frac{42\!\cdots\!47}{74\!\cdots\!75}a^{10}-\frac{47\!\cdots\!24}{45\!\cdots\!75}a^{9}+\frac{44\!\cdots\!57}{45\!\cdots\!75}a^{8}-\frac{12\!\cdots\!43}{45\!\cdots\!75}a^{7}-\frac{36\!\cdots\!91}{90\!\cdots\!75}a^{6}-\frac{32\!\cdots\!69}{90\!\cdots\!75}a^{5}+\frac{87\!\cdots\!37}{18\!\cdots\!75}a^{4}+\frac{25\!\cdots\!87}{18\!\cdots\!75}a^{3}+\frac{27\!\cdots\!47}{15\!\cdots\!85}a^{2}+\frac{56\!\cdots\!88}{54\!\cdots\!65}a+\frac{12\!\cdots\!85}{10\!\cdots\!33}$ 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:  $5$

Class group and class number

$C_{3}$, which has order $3$ (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:   $\frac{65\!\cdots\!28}{10\!\cdots\!75}a^{26}-\frac{41\!\cdots\!68}{10\!\cdots\!75}a^{25}+\frac{50\!\cdots\!49}{10\!\cdots\!75}a^{24}+\frac{47\!\cdots\!83}{98\!\cdots\!25}a^{23}-\frac{12\!\cdots\!77}{10\!\cdots\!75}a^{22}-\frac{99\!\cdots\!37}{47\!\cdots\!25}a^{21}+\frac{29\!\cdots\!56}{27\!\cdots\!25}a^{20}-\frac{56\!\cdots\!38}{10\!\cdots\!75}a^{19}+\frac{82\!\cdots\!44}{98\!\cdots\!25}a^{18}+\frac{11\!\cdots\!14}{10\!\cdots\!75}a^{17}-\frac{36\!\cdots\!73}{98\!\cdots\!25}a^{16}+\frac{17\!\cdots\!93}{10\!\cdots\!75}a^{15}+\frac{25\!\cdots\!71}{10\!\cdots\!75}a^{14}-\frac{14\!\cdots\!93}{10\!\cdots\!75}a^{13}+\frac{30\!\cdots\!92}{10\!\cdots\!75}a^{12}+\frac{30\!\cdots\!66}{10\!\cdots\!75}a^{11}-\frac{62\!\cdots\!81}{17\!\cdots\!75}a^{10}-\frac{10\!\cdots\!62}{10\!\cdots\!75}a^{9}+\frac{17\!\cdots\!06}{10\!\cdots\!75}a^{8}+\frac{21\!\cdots\!11}{10\!\cdots\!75}a^{7}+\frac{98\!\cdots\!89}{21\!\cdots\!75}a^{6}-\frac{26\!\cdots\!32}{21\!\cdots\!75}a^{5}+\frac{15\!\cdots\!37}{17\!\cdots\!43}a^{4}+\frac{47\!\cdots\!06}{43\!\cdots\!75}a^{3}+\frac{32\!\cdots\!14}{37\!\cdots\!05}a^{2}-\frac{27\!\cdots\!41}{11\!\cdots\!95}a+\frac{26\!\cdots\!46}{25\!\cdots\!29}$, $\frac{14\!\cdots\!58}{10\!\cdots\!75}a^{26}-\frac{14\!\cdots\!77}{10\!\cdots\!75}a^{25}+\frac{37\!\cdots\!76}{66\!\cdots\!25}a^{24}+\frac{11\!\cdots\!08}{63\!\cdots\!75}a^{23}-\frac{98\!\cdots\!06}{10\!\cdots\!75}a^{22}+\frac{40\!\cdots\!18}{18\!\cdots\!25}a^{21}+\frac{10\!\cdots\!38}{47\!\cdots\!25}a^{20}-\frac{25\!\cdots\!21}{10\!\cdots\!75}a^{19}+\frac{97\!\cdots\!33}{10\!\cdots\!75}a^{18}-\frac{17\!\cdots\!08}{10\!\cdots\!75}a^{17}-\frac{40\!\cdots\!27}{21\!\cdots\!75}a^{16}+\frac{80\!\cdots\!27}{10\!\cdots\!75}a^{15}+\frac{83\!\cdots\!57}{10\!\cdots\!75}a^{14}-\frac{60\!\cdots\!26}{10\!\cdots\!75}a^{13}+\frac{25\!\cdots\!11}{10\!\cdots\!75}a^{12}-\frac{12\!\cdots\!86}{21\!\cdots\!75}a^{11}+\frac{44\!\cdots\!51}{17\!\cdots\!75}a^{10}+\frac{80\!\cdots\!31}{10\!\cdots\!75}a^{9}+\frac{30\!\cdots\!82}{10\!\cdots\!75}a^{8}-\frac{42\!\cdots\!62}{10\!\cdots\!75}a^{7}-\frac{33\!\cdots\!78}{10\!\cdots\!75}a^{6}-\frac{25\!\cdots\!34}{21\!\cdots\!75}a^{5}+\frac{63\!\cdots\!16}{21\!\cdots\!75}a^{4}+\frac{42\!\cdots\!64}{43\!\cdots\!75}a^{3}-\frac{14\!\cdots\!21}{18\!\cdots\!25}a^{2}-\frac{63\!\cdots\!48}{12\!\cdots\!45}a+\frac{42\!\cdots\!78}{11\!\cdots\!95}$, $\frac{10\!\cdots\!43}{10\!\cdots\!75}a^{26}-\frac{41\!\cdots\!22}{10\!\cdots\!75}a^{25}-\frac{17\!\cdots\!17}{10\!\cdots\!75}a^{24}+\frac{85\!\cdots\!61}{10\!\cdots\!75}a^{23}+\frac{40\!\cdots\!99}{10\!\cdots\!75}a^{22}-\frac{42\!\cdots\!91}{94\!\cdots\!25}a^{21}+\frac{62\!\cdots\!48}{47\!\cdots\!25}a^{20}-\frac{48\!\cdots\!86}{10\!\cdots\!75}a^{19}-\frac{71\!\cdots\!67}{10\!\cdots\!75}a^{18}+\frac{49\!\cdots\!37}{98\!\cdots\!25}a^{17}-\frac{74\!\cdots\!21}{21\!\cdots\!75}a^{16}-\frac{10\!\cdots\!08}{10\!\cdots\!75}a^{15}+\frac{14\!\cdots\!42}{98\!\cdots\!25}a^{14}-\frac{15\!\cdots\!51}{10\!\cdots\!75}a^{13}-\frac{68\!\cdots\!94}{10\!\cdots\!75}a^{12}+\frac{26\!\cdots\!07}{21\!\cdots\!75}a^{11}+\frac{49\!\cdots\!46}{17\!\cdots\!75}a^{10}-\frac{19\!\cdots\!54}{10\!\cdots\!75}a^{9}-\frac{85\!\cdots\!93}{10\!\cdots\!75}a^{8}+\frac{42\!\cdots\!23}{10\!\cdots\!75}a^{7}+\frac{22\!\cdots\!07}{10\!\cdots\!75}a^{6}+\frac{59\!\cdots\!72}{21\!\cdots\!75}a^{5}-\frac{63\!\cdots\!22}{11\!\cdots\!25}a^{4}-\frac{20\!\cdots\!06}{43\!\cdots\!75}a^{3}-\frac{12\!\cdots\!51}{18\!\cdots\!25}a^{2}+\frac{37\!\cdots\!81}{25\!\cdots\!29}a+\frac{71\!\cdots\!63}{12\!\cdots\!45}$, $\frac{16\!\cdots\!17}{25\!\cdots\!75}a^{26}-\frac{49\!\cdots\!46}{10\!\cdots\!75}a^{25}+\frac{27\!\cdots\!01}{10\!\cdots\!75}a^{24}+\frac{59\!\cdots\!52}{10\!\cdots\!75}a^{23}-\frac{16\!\cdots\!51}{10\!\cdots\!75}a^{22}+\frac{54\!\cdots\!58}{47\!\cdots\!25}a^{21}+\frac{57\!\cdots\!64}{47\!\cdots\!25}a^{20}-\frac{67\!\cdots\!22}{10\!\cdots\!75}a^{19}+\frac{12\!\cdots\!01}{10\!\cdots\!75}a^{18}+\frac{10\!\cdots\!07}{10\!\cdots\!75}a^{17}-\frac{54\!\cdots\!38}{10\!\cdots\!75}a^{16}+\frac{13\!\cdots\!91}{10\!\cdots\!75}a^{15}+\frac{29\!\cdots\!99}{10\!\cdots\!75}a^{14}-\frac{17\!\cdots\!17}{10\!\cdots\!75}a^{13}+\frac{40\!\cdots\!86}{10\!\cdots\!75}a^{12}-\frac{33\!\cdots\!17}{63\!\cdots\!75}a^{11}-\frac{95\!\cdots\!57}{17\!\cdots\!75}a^{10}-\frac{10\!\cdots\!13}{10\!\cdots\!75}a^{9}+\frac{27\!\cdots\!24}{10\!\cdots\!75}a^{8}+\frac{18\!\cdots\!33}{98\!\cdots\!25}a^{7}+\frac{48\!\cdots\!38}{10\!\cdots\!75}a^{6}-\frac{33\!\cdots\!67}{19\!\cdots\!25}a^{5}+\frac{47\!\cdots\!92}{11\!\cdots\!25}a^{4}+\frac{72\!\cdots\!27}{43\!\cdots\!75}a^{3}+\frac{16\!\cdots\!11}{18\!\cdots\!25}a^{2}-\frac{88\!\cdots\!99}{25\!\cdots\!29}a+\frac{19\!\cdots\!07}{12\!\cdots\!45}$, $\frac{33\!\cdots\!81}{45\!\cdots\!75}a^{26}-\frac{46\!\cdots\!77}{90\!\cdots\!75}a^{25}+\frac{10\!\cdots\!82}{45\!\cdots\!75}a^{24}+\frac{27\!\cdots\!49}{45\!\cdots\!75}a^{23}-\frac{75\!\cdots\!68}{45\!\cdots\!75}a^{22}+\frac{27\!\cdots\!13}{19\!\cdots\!25}a^{21}+\frac{53\!\cdots\!26}{39\!\cdots\!25}a^{20}-\frac{31\!\cdots\!24}{45\!\cdots\!75}a^{19}+\frac{57\!\cdots\!67}{45\!\cdots\!75}a^{18}+\frac{47\!\cdots\!01}{45\!\cdots\!75}a^{17}-\frac{24\!\cdots\!88}{45\!\cdots\!75}a^{16}+\frac{79\!\cdots\!48}{53\!\cdots\!75}a^{15}+\frac{13\!\cdots\!58}{45\!\cdots\!75}a^{14}-\frac{81\!\cdots\!19}{45\!\cdots\!75}a^{13}+\frac{18\!\cdots\!13}{45\!\cdots\!75}a^{12}-\frac{25\!\cdots\!74}{41\!\cdots\!25}a^{11}-\frac{15\!\cdots\!07}{29\!\cdots\!75}a^{10}-\frac{44\!\cdots\!31}{41\!\cdots\!25}a^{9}+\frac{11\!\cdots\!13}{45\!\cdots\!75}a^{8}-\frac{46\!\cdots\!21}{45\!\cdots\!75}a^{7}+\frac{23\!\cdots\!22}{45\!\cdots\!75}a^{6}-\frac{64\!\cdots\!22}{36\!\cdots\!55}a^{5}+\frac{47\!\cdots\!46}{90\!\cdots\!75}a^{4}+\frac{59\!\cdots\!39}{36\!\cdots\!55}a^{3}+\frac{52\!\cdots\!29}{71\!\cdots\!75}a^{2}-\frac{19\!\cdots\!11}{54\!\cdots\!65}a+\frac{10\!\cdots\!08}{54\!\cdots\!65}$, $\frac{23\!\cdots\!11}{45\!\cdots\!75}a^{26}-\frac{15\!\cdots\!57}{45\!\cdots\!75}a^{25}+\frac{67\!\cdots\!39}{41\!\cdots\!25}a^{24}+\frac{19\!\cdots\!13}{45\!\cdots\!75}a^{23}-\frac{10\!\cdots\!01}{90\!\cdots\!75}a^{22}+\frac{17\!\cdots\!14}{19\!\cdots\!25}a^{21}+\frac{16\!\cdots\!73}{17\!\cdots\!75}a^{20}-\frac{21\!\cdots\!08}{45\!\cdots\!75}a^{19}+\frac{39\!\cdots\!24}{45\!\cdots\!75}a^{18}+\frac{38\!\cdots\!26}{53\!\cdots\!75}a^{17}-\frac{16\!\cdots\!74}{45\!\cdots\!75}a^{16}+\frac{45\!\cdots\!42}{45\!\cdots\!75}a^{15}+\frac{93\!\cdots\!86}{45\!\cdots\!75}a^{14}-\frac{56\!\cdots\!88}{45\!\cdots\!75}a^{13}+\frac{25\!\cdots\!04}{90\!\cdots\!75}a^{12}-\frac{16\!\cdots\!27}{41\!\cdots\!25}a^{11}-\frac{26\!\cdots\!99}{74\!\cdots\!75}a^{10}-\frac{33\!\cdots\!02}{45\!\cdots\!75}a^{9}+\frac{47\!\cdots\!43}{26\!\cdots\!75}a^{8}-\frac{56\!\cdots\!88}{90\!\cdots\!75}a^{7}+\frac{16\!\cdots\!28}{45\!\cdots\!75}a^{6}-\frac{10\!\cdots\!29}{90\!\cdots\!75}a^{5}+\frac{29\!\cdots\!84}{82\!\cdots\!25}a^{4}+\frac{20\!\cdots\!34}{18\!\cdots\!75}a^{3}+\frac{40\!\cdots\!06}{78\!\cdots\!25}a^{2}-\frac{13\!\cdots\!21}{54\!\cdots\!65}a+\frac{65\!\cdots\!17}{49\!\cdots\!15}$, $\frac{43\!\cdots\!36}{45\!\cdots\!75}a^{26}-\frac{23\!\cdots\!79}{45\!\cdots\!75}a^{25}-\frac{10\!\cdots\!37}{26\!\cdots\!75}a^{24}+\frac{46\!\cdots\!61}{67\!\cdots\!25}a^{23}-\frac{45\!\cdots\!33}{39\!\cdots\!25}a^{22}-\frac{35\!\cdots\!01}{39\!\cdots\!25}a^{21}+\frac{29\!\cdots\!66}{19\!\cdots\!25}a^{20}-\frac{28\!\cdots\!37}{41\!\cdots\!25}a^{19}+\frac{19\!\cdots\!83}{26\!\cdots\!75}a^{18}+\frac{84\!\cdots\!09}{45\!\cdots\!75}a^{17}-\frac{31\!\cdots\!28}{90\!\cdots\!75}a^{16}-\frac{65\!\cdots\!78}{61\!\cdots\!75}a^{15}+\frac{15\!\cdots\!19}{45\!\cdots\!75}a^{14}-\frac{83\!\cdots\!67}{45\!\cdots\!75}a^{13}+\frac{12\!\cdots\!47}{45\!\cdots\!75}a^{12}+\frac{16\!\cdots\!86}{90\!\cdots\!75}a^{11}-\frac{13\!\cdots\!93}{74\!\cdots\!75}a^{10}-\frac{60\!\cdots\!23}{45\!\cdots\!75}a^{9}+\frac{49\!\cdots\!69}{45\!\cdots\!75}a^{8}-\frac{74\!\cdots\!49}{45\!\cdots\!75}a^{7}+\frac{31\!\cdots\!19}{45\!\cdots\!75}a^{6}-\frac{20\!\cdots\!17}{18\!\cdots\!75}a^{5}-\frac{23\!\cdots\!13}{82\!\cdots\!25}a^{4}+\frac{14\!\cdots\!03}{18\!\cdots\!75}a^{3}+\frac{10\!\cdots\!78}{78\!\cdots\!25}a^{2}-\frac{11\!\cdots\!12}{54\!\cdots\!65}a+\frac{45\!\cdots\!01}{54\!\cdots\!65}$, $\frac{99\!\cdots\!08}{53\!\cdots\!75}a^{26}-\frac{11\!\cdots\!62}{90\!\cdots\!75}a^{25}+\frac{11\!\cdots\!23}{18\!\cdots\!75}a^{24}+\frac{13\!\cdots\!52}{90\!\cdots\!75}a^{23}-\frac{37\!\cdots\!02}{90\!\cdots\!75}a^{22}+\frac{14\!\cdots\!16}{39\!\cdots\!25}a^{21}+\frac{15\!\cdots\!91}{46\!\cdots\!25}a^{20}-\frac{31\!\cdots\!59}{18\!\cdots\!75}a^{19}+\frac{29\!\cdots\!91}{90\!\cdots\!75}a^{18}+\frac{13\!\cdots\!32}{53\!\cdots\!75}a^{17}-\frac{72\!\cdots\!93}{53\!\cdots\!75}a^{16}+\frac{68\!\cdots\!37}{18\!\cdots\!75}a^{15}+\frac{13\!\cdots\!18}{18\!\cdots\!75}a^{14}-\frac{41\!\cdots\!32}{90\!\cdots\!75}a^{13}+\frac{93\!\cdots\!32}{90\!\cdots\!75}a^{12}-\frac{75\!\cdots\!24}{48\!\cdots\!25}a^{11}-\frac{39\!\cdots\!54}{29\!\cdots\!75}a^{10}-\frac{49\!\cdots\!49}{18\!\cdots\!75}a^{9}+\frac{59\!\cdots\!49}{90\!\cdots\!75}a^{8}-\frac{25\!\cdots\!09}{90\!\cdots\!75}a^{7}+\frac{23\!\cdots\!83}{18\!\cdots\!75}a^{6}-\frac{40\!\cdots\!68}{90\!\cdots\!75}a^{5}+\frac{24\!\cdots\!89}{18\!\cdots\!75}a^{4}+\frac{14\!\cdots\!19}{36\!\cdots\!55}a^{3}+\frac{29\!\cdots\!88}{15\!\cdots\!85}a^{2}-\frac{48\!\cdots\!04}{54\!\cdots\!65}a+\frac{52\!\cdots\!69}{10\!\cdots\!33}$, $\frac{68\!\cdots\!61}{45\!\cdots\!75}a^{26}-\frac{26\!\cdots\!01}{45\!\cdots\!75}a^{25}-\frac{78\!\cdots\!67}{45\!\cdots\!75}a^{24}+\frac{42\!\cdots\!26}{45\!\cdots\!75}a^{23}-\frac{14\!\cdots\!79}{45\!\cdots\!75}a^{22}-\frac{67\!\cdots\!09}{19\!\cdots\!25}a^{21}+\frac{40\!\cdots\!99}{17\!\cdots\!75}a^{20}-\frac{33\!\cdots\!96}{45\!\cdots\!75}a^{19}-\frac{16\!\cdots\!32}{45\!\cdots\!75}a^{18}+\frac{18\!\cdots\!03}{45\!\cdots\!75}a^{17}-\frac{71\!\cdots\!21}{45\!\cdots\!75}a^{16}-\frac{20\!\cdots\!22}{26\!\cdots\!75}a^{15}+\frac{25\!\cdots\!32}{45\!\cdots\!75}a^{14}-\frac{10\!\cdots\!61}{45\!\cdots\!75}a^{13}+\frac{15\!\cdots\!19}{41\!\cdots\!25}a^{12}+\frac{34\!\cdots\!37}{45\!\cdots\!75}a^{11}+\frac{51\!\cdots\!58}{74\!\cdots\!75}a^{10}-\frac{97\!\cdots\!29}{45\!\cdots\!75}a^{9}+\frac{72\!\cdots\!37}{45\!\cdots\!75}a^{8}-\frac{68\!\cdots\!53}{45\!\cdots\!75}a^{7}+\frac{45\!\cdots\!34}{36\!\cdots\!55}a^{6}-\frac{59\!\cdots\!09}{90\!\cdots\!75}a^{5}-\frac{29\!\cdots\!08}{18\!\cdots\!75}a^{4}+\frac{95\!\cdots\!62}{18\!\cdots\!75}a^{3}+\frac{34\!\cdots\!02}{14\!\cdots\!35}a^{2}-\frac{10\!\cdots\!72}{54\!\cdots\!65}a+\frac{34\!\cdots\!68}{10\!\cdots\!33}$, $\frac{66\!\cdots\!66}{41\!\cdots\!25}a^{26}-\frac{49\!\cdots\!47}{45\!\cdots\!75}a^{25}+\frac{25\!\cdots\!79}{45\!\cdots\!75}a^{24}+\frac{58\!\cdots\!78}{45\!\cdots\!75}a^{23}-\frac{65\!\cdots\!97}{18\!\cdots\!75}a^{22}+\frac{14\!\cdots\!94}{19\!\cdots\!25}a^{21}+\frac{57\!\cdots\!63}{19\!\cdots\!25}a^{20}-\frac{68\!\cdots\!83}{45\!\cdots\!75}a^{19}+\frac{12\!\cdots\!44}{45\!\cdots\!75}a^{18}+\frac{33\!\cdots\!74}{18\!\cdots\!75}a^{17}-\frac{52\!\cdots\!79}{45\!\cdots\!75}a^{16}+\frac{22\!\cdots\!32}{45\!\cdots\!75}a^{15}+\frac{29\!\cdots\!61}{45\!\cdots\!75}a^{14}-\frac{17\!\cdots\!78}{45\!\cdots\!75}a^{13}+\frac{81\!\cdots\!78}{90\!\cdots\!75}a^{12}-\frac{11\!\cdots\!37}{45\!\cdots\!75}a^{11}-\frac{82\!\cdots\!29}{74\!\cdots\!75}a^{10}-\frac{89\!\cdots\!52}{45\!\cdots\!75}a^{9}+\frac{26\!\cdots\!86}{45\!\cdots\!75}a^{8}-\frac{10\!\cdots\!66}{90\!\cdots\!75}a^{7}+\frac{41\!\cdots\!38}{41\!\cdots\!25}a^{6}-\frac{20\!\cdots\!67}{53\!\cdots\!75}a^{5}+\frac{89\!\cdots\!72}{53\!\cdots\!75}a^{4}+\frac{66\!\cdots\!64}{18\!\cdots\!75}a^{3}+\frac{74\!\cdots\!76}{78\!\cdots\!25}a^{2}-\frac{44\!\cdots\!96}{54\!\cdots\!65}a+\frac{25\!\cdots\!62}{54\!\cdots\!65}$, $\frac{15\!\cdots\!01}{41\!\cdots\!25}a^{26}-\frac{11\!\cdots\!39}{45\!\cdots\!75}a^{25}+\frac{24\!\cdots\!11}{41\!\cdots\!25}a^{24}+\frac{14\!\cdots\!82}{45\!\cdots\!75}a^{23}-\frac{35\!\cdots\!32}{45\!\cdots\!75}a^{22}-\frac{51\!\cdots\!09}{39\!\cdots\!25}a^{21}+\frac{13\!\cdots\!91}{19\!\cdots\!25}a^{20}-\frac{15\!\cdots\!07}{45\!\cdots\!75}a^{19}+\frac{25\!\cdots\!21}{45\!\cdots\!75}a^{18}+\frac{30\!\cdots\!74}{45\!\cdots\!75}a^{17}-\frac{46\!\cdots\!16}{18\!\cdots\!75}a^{16}+\frac{22\!\cdots\!39}{45\!\cdots\!75}a^{15}+\frac{69\!\cdots\!69}{45\!\cdots\!75}a^{14}-\frac{14\!\cdots\!47}{16\!\cdots\!25}a^{13}+\frac{85\!\cdots\!67}{45\!\cdots\!75}a^{12}+\frac{30\!\cdots\!87}{18\!\cdots\!75}a^{11}-\frac{17\!\cdots\!38}{67\!\cdots\!25}a^{10}-\frac{29\!\cdots\!98}{45\!\cdots\!75}a^{9}+\frac{51\!\cdots\!84}{45\!\cdots\!75}a^{8}+\frac{12\!\cdots\!61}{45\!\cdots\!75}a^{7}+\frac{13\!\cdots\!29}{45\!\cdots\!75}a^{6}-\frac{75\!\cdots\!18}{90\!\cdots\!75}a^{5}+\frac{32\!\cdots\!37}{90\!\cdots\!75}a^{4}+\frac{14\!\cdots\!43}{18\!\cdots\!75}a^{3}+\frac{50\!\cdots\!03}{78\!\cdots\!25}a^{2}-\frac{85\!\cdots\!86}{54\!\cdots\!65}a+\frac{31\!\cdots\!51}{54\!\cdots\!65}$, $\frac{42\!\cdots\!43}{45\!\cdots\!75}a^{26}+\frac{10\!\cdots\!69}{45\!\cdots\!75}a^{25}-\frac{74\!\cdots\!83}{45\!\cdots\!75}a^{24}-\frac{12\!\cdots\!96}{45\!\cdots\!75}a^{23}+\frac{19\!\cdots\!09}{90\!\cdots\!75}a^{22}-\frac{76\!\cdots\!18}{19\!\cdots\!25}a^{21}-\frac{59\!\cdots\!26}{17\!\cdots\!75}a^{20}+\frac{18\!\cdots\!16}{45\!\cdots\!75}a^{19}-\frac{90\!\cdots\!58}{45\!\cdots\!75}a^{18}+\frac{12\!\cdots\!96}{48\!\cdots\!25}a^{17}+\frac{32\!\cdots\!13}{45\!\cdots\!75}a^{16}-\frac{59\!\cdots\!54}{45\!\cdots\!75}a^{15}-\frac{48\!\cdots\!47}{45\!\cdots\!75}a^{14}+\frac{37\!\cdots\!96}{45\!\cdots\!75}a^{13}-\frac{19\!\cdots\!02}{36\!\cdots\!55}a^{12}+\frac{42\!\cdots\!64}{45\!\cdots\!75}a^{11}+\frac{76\!\cdots\!13}{74\!\cdots\!75}a^{10}-\frac{60\!\cdots\!71}{45\!\cdots\!75}a^{9}-\frac{28\!\cdots\!27}{45\!\cdots\!75}a^{8}+\frac{46\!\cdots\!48}{18\!\cdots\!75}a^{7}+\frac{32\!\cdots\!14}{45\!\cdots\!75}a^{6}+\frac{15\!\cdots\!83}{53\!\cdots\!75}a^{5}-\frac{31\!\cdots\!93}{90\!\cdots\!75}a^{4}-\frac{67\!\cdots\!53}{18\!\cdots\!75}a^{3}+\frac{14\!\cdots\!98}{78\!\cdots\!25}a^{2}+\frac{11\!\cdots\!54}{17\!\cdots\!85}a-\frac{22\!\cdots\!44}{49\!\cdots\!15}$, $\frac{78\!\cdots\!49}{90\!\cdots\!75}a^{26}-\frac{30\!\cdots\!57}{45\!\cdots\!75}a^{25}+\frac{31\!\cdots\!17}{45\!\cdots\!75}a^{24}+\frac{30\!\cdots\!64}{41\!\cdots\!25}a^{23}-\frac{11\!\cdots\!22}{45\!\cdots\!75}a^{22}+\frac{22\!\cdots\!41}{19\!\cdots\!25}a^{21}+\frac{33\!\cdots\!33}{19\!\cdots\!25}a^{20}-\frac{42\!\cdots\!99}{45\!\cdots\!75}a^{19}+\frac{91\!\cdots\!02}{45\!\cdots\!75}a^{18}+\frac{24\!\cdots\!89}{41\!\cdots\!25}a^{17}-\frac{36\!\cdots\!51}{45\!\cdots\!75}a^{16}+\frac{22\!\cdots\!77}{45\!\cdots\!75}a^{15}+\frac{16\!\cdots\!08}{45\!\cdots\!75}a^{14}-\frac{63\!\cdots\!02}{26\!\cdots\!75}a^{13}+\frac{16\!\cdots\!76}{26\!\cdots\!75}a^{12}-\frac{13\!\cdots\!28}{45\!\cdots\!75}a^{11}-\frac{60\!\cdots\!79}{74\!\cdots\!75}a^{10}-\frac{44\!\cdots\!71}{45\!\cdots\!75}a^{9}+\frac{19\!\cdots\!98}{45\!\cdots\!75}a^{8}-\frac{51\!\cdots\!14}{45\!\cdots\!75}a^{7}+\frac{23\!\cdots\!06}{45\!\cdots\!75}a^{6}-\frac{48\!\cdots\!56}{18\!\cdots\!75}a^{5}+\frac{14\!\cdots\!33}{90\!\cdots\!75}a^{4}+\frac{46\!\cdots\!39}{18\!\cdots\!75}a^{3}-\frac{42\!\cdots\!43}{78\!\cdots\!25}a^{2}-\frac{28\!\cdots\!88}{49\!\cdots\!15}a+\frac{21\!\cdots\!69}{54\!\cdots\!65}$ 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:  \( 71364448730505.47 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 71364448730505.47 \cdot 3}{2\cdot\sqrt{1089801024238603052304820653163822019730224609375}}\cr\approx \mathstrut & 4.87829267984210 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625)
 
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 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625, 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 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625);
 
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 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625);
 
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

3.1.335.1, 9.1.12594450625.1

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 $27$ $27$ R R ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.9.0.1}{9} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.9.0.1}{9} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $27$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $27$ ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $27$ $27$

In the table, R denotes a ramified prime. 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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
\(7\) Copy content Toggle raw display Deg $27$$3$$9$$18$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.335.2t1.a.a$1$ $ 5 \cdot 67 $ \(\Q(\sqrt{-335}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.335.3t2.a.a$2$ $ 5 \cdot 67 $ 3.1.335.1 $S_3$ (as 3T2) $1$ $0$
* 2.335.9t3.a.b$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.c$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.a$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.16415.27t8.a.e$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.f$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.a$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.h$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.c$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.i$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.g$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.d$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.b$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (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 *.