Properties

Label 27.1.149...032.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.497\times 10^{41}$
Root discriminant $33.50$
Ramified primes $2, 563$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80)
 
gp: K = bnfinit(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-80, 848, -4224, 13340, -29876, 50256, -65648, 70628, -72912, 89765, -127135, 169389, -189870, 176010, -138060, 93410, -54584, 28791, -13591, 5743, -2140, 734, -244, 110, -50, 21, -7, 1]);
 

\( x^{27} - 7 x^{26} + 21 x^{25} - 50 x^{24} + 110 x^{23} - 244 x^{22} + 734 x^{21} - 2140 x^{20} + 5743 x^{19} - 13591 x^{18} + 28791 x^{17} - 54584 x^{16} + 93410 x^{15} - 138060 x^{14} + 176010 x^{13} - 189870 x^{12} + 169389 x^{11} - 127135 x^{10} + 89765 x^{9} - 72912 x^{8} + 70628 x^{7} - 65648 x^{6} + 50256 x^{5} - 29876 x^{4} + 13340 x^{3} - 4224 x^{2} + 848 x - 80 \)

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-149674927005884133619412112407487159468032\)\(\medspace = -\,2^{18}\cdot 563^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.50$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 563$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{17} + \frac{1}{10} a^{16} - \frac{1}{20} a^{15} - \frac{1}{10} a^{14} + \frac{1}{5} a^{13} - \frac{1}{10} a^{12} - \frac{1}{20} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{8} - \frac{1}{4} a^{7} - \frac{1}{20} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{2} - \frac{2}{5} a$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{14} - \frac{1}{4} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{3}{20} a^{9} - \frac{3}{10} a^{8} - \frac{1}{20} a^{6} + \frac{9}{20} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} - \frac{1}{5} a$, $\frac{1}{20} a^{19} - \frac{1}{10} a^{15} - \frac{1}{4} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{3}{20} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{1}{20} a^{7} - \frac{1}{20} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{40} a^{20} - \frac{1}{40} a^{19} - \frac{1}{40} a^{17} + \frac{1}{40} a^{16} - \frac{1}{20} a^{15} + \frac{1}{8} a^{14} - \frac{9}{40} a^{13} - \frac{3}{40} a^{12} - \frac{1}{10} a^{11} + \frac{1}{8} a^{10} - \frac{9}{40} a^{9} - \frac{1}{10} a^{8} + \frac{1}{8} a^{7} + \frac{17}{40} a^{6} + \frac{3}{10} a^{5} + \frac{1}{10} a^{4} + \frac{9}{20} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5} a$, $\frac{1}{40} a^{21} - \frac{1}{40} a^{19} - \frac{1}{40} a^{18} - \frac{1}{40} a^{16} + \frac{3}{40} a^{15} - \frac{1}{10} a^{14} + \frac{1}{5} a^{13} - \frac{7}{40} a^{12} + \frac{1}{40} a^{11} - \frac{1}{10} a^{10} + \frac{7}{40} a^{9} + \frac{1}{40} a^{8} - \frac{9}{20} a^{7} - \frac{11}{40} a^{6} - \frac{1}{10} a^{5} - \frac{9}{20} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a$, $\frac{1}{40} a^{22} - \frac{1}{20} a^{16} - \frac{1}{20} a^{15} + \frac{9}{40} a^{14} + \frac{1}{5} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{19}{40} a^{6} + \frac{1}{10} a^{5} - \frac{9}{20} a^{4} - \frac{3}{20} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{40} a^{23} + \frac{1}{20} a^{16} - \frac{3}{40} a^{15} - \frac{3}{20} a^{14} - \frac{1}{20} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{20} a^{10} - \frac{3}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{40} a^{7} - \frac{3}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{20} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{40} a^{24} + \frac{3}{40} a^{16} - \frac{1}{10} a^{15} + \frac{1}{20} a^{14} - \frac{3}{20} a^{13} + \frac{1}{20} a^{12} + \frac{1}{10} a^{11} + \frac{1}{20} a^{10} - \frac{1}{20} a^{9} + \frac{3}{8} a^{8} - \frac{2}{5} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{973400} a^{25} - \frac{137}{97340} a^{24} - \frac{737}{97340} a^{23} - \frac{4}{785} a^{22} + \frac{489}{97340} a^{21} - \frac{5739}{973400} a^{20} - \frac{14249}{973400} a^{19} + \frac{2012}{121675} a^{18} + \frac{77}{15700} a^{17} + \frac{621}{31400} a^{16} - \frac{4869}{243350} a^{15} - \frac{173167}{973400} a^{14} + \frac{4707}{31400} a^{13} + \frac{160601}{973400} a^{12} + \frac{277}{24335} a^{11} - \frac{209751}{973400} a^{10} + \frac{27823}{243350} a^{9} - \frac{2227}{19468} a^{8} + \frac{120443}{973400} a^{7} - \frac{325971}{973400} a^{6} - \frac{26279}{121675} a^{5} + \frac{2821}{7850} a^{4} + \frac{177613}{486700} a^{3} - \frac{40787}{243350} a^{2} - \frac{56989}{121675} a + \frac{5748}{24335}$, $\frac{1}{36780145240738430396201552959538777000} a^{26} - \frac{71229904979266566300430724512}{148307037261042058049199810320720875} a^{25} - \frac{55930775430782559388279940543055987}{7356029048147686079240310591907755400} a^{24} - \frac{6902941856408461776965683635737411}{3678014524073843039620155295953877700} a^{23} - \frac{7903911116310811080278450687257691}{1471205809629537215848062118381551080} a^{22} - \frac{50023599997166183298482387857148968}{4597518155092303799525194119942347125} a^{21} + \frac{20208106414630741400362409024458309}{7356029048147686079240310591907755400} a^{20} + \frac{120821099224319928917148331018931001}{7356029048147686079240310591907755400} a^{19} + \frac{318495498524937405890390504192716723}{36780145240738430396201552959538777000} a^{18} + \frac{11840379126968696074718097017070587}{1186456298088336464393598482565767000} a^{17} - \frac{227749930802853890200765610549494801}{18390072620369215198100776479769388500} a^{16} + \frac{1807485432875035154488869825775803077}{18390072620369215198100776479769388500} a^{15} - \frac{565695241699780992525278367374063729}{9195036310184607599050388239884694250} a^{14} + \frac{4262525231837336397756830574368420219}{36780145240738430396201552959538777000} a^{13} + \frac{5075644638805319627808081934587175999}{36780145240738430396201552959538777000} a^{12} + \frac{748359368691465578433331388763999681}{9195036310184607599050388239884694250} a^{11} + \frac{945872447522729370973144930221996326}{4597518155092303799525194119942347125} a^{10} + \frac{4700402157486075254911752090656895463}{36780145240738430396201552959538777000} a^{9} - \frac{9532466378859860213929350944933769957}{36780145240738430396201552959538777000} a^{8} - \frac{4475202718578510749523357325657256279}{36780145240738430396201552959538777000} a^{7} + \frac{2121727607180706448400995205902423277}{18390072620369215198100776479769388500} a^{6} - \frac{6212173002965935698759031628099623087}{18390072620369215198100776479769388500} a^{5} - \frac{4392006291370127650358930641832926169}{18390072620369215198100776479769388500} a^{4} - \frac{1264526551414922658515745850498254651}{9195036310184607599050388239884694250} a^{3} - \frac{3152349305454504139724675129548697371}{9195036310184607599050388239884694250} a^{2} - \frac{2137599926502749520296877232776359216}{4597518155092303799525194119942347125} a - \frac{445885343244733984234298607402289833}{919503631018460759905038823988469425}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 14887194175.434155 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 14887194175.434155 \cdot 2}{2\sqrt{149674927005884133619412112407487159468032}}\approx 1.83065543691759$ (assuming GRH)

Galois group

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

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

Intermediate fields

3.1.563.1, 9.1.100469346961.1

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 R $27$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
563Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.563.2t1.a.a$1$ $ 563 $ \(\Q(\sqrt{-563}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.563.3t2.a.a$2$ $ 563 $ 3.1.563.1 $S_3$ (as 3T2) $1$ $0$
* 2.563.9t3.a.c$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.a$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.b$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2252.27t8.a.c$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.b$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.e$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.i$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.f$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.a$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.h$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.g$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.d$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (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 *.