Properties

Label 27.1.201...127.1
Degree $27$
Signature $[1, 13]$
Discriminant $-2.019\times 10^{46}$
Root discriminant $51.88$
Ramified primes $7, 521$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 43*x^25 - 96*x^24 + 144*x^23 - 267*x^22 + 429*x^21 + 20*x^20 - 947*x^19 + 1263*x^18 + 1070*x^17 - 4443*x^16 + 3518*x^15 + 4729*x^14 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075)
 
gp: K = bnfinit(x^27 - 9*x^26 + 43*x^25 - 96*x^24 + 144*x^23 - 267*x^22 + 429*x^21 + 20*x^20 - 947*x^19 + 1263*x^18 + 1070*x^17 - 4443*x^16 + 3518*x^15 + 4729*x^14 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2075, 24633, -46897, 900, 47488, 6811, -14482, 17843, 6002, -8600, -1996, 13136, 2497, -7874, 4729, 3518, -4443, 1070, 1263, -947, 20, 429, -267, 144, -96, 43, -9, 1]);
 

\( x^{27} - 9 x^{26} + 43 x^{25} - 96 x^{24} + 144 x^{23} - 267 x^{22} + 429 x^{21} + 20 x^{20} - 947 x^{19} + 1263 x^{18} + 1070 x^{17} - 4443 x^{16} + 3518 x^{15} + 4729 x^{14} - 7874 x^{13} + 2497 x^{12} + 13136 x^{11} - 1996 x^{10} - 8600 x^{9} + 6002 x^{8} + 17843 x^{7} - 14482 x^{6} + 6811 x^{5} + 47488 x^{4} + 900 x^{3} - 46897 x^{2} + 24633 x + 2075 \)

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:  \(-20191329826970487018697554420683199956085496127\)\(\medspace = -\,7^{13}\cdot 521^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.88$
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:  $7, 521$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35} a^{11} - \frac{13}{35} a^{10} - \frac{13}{35} a^{9} - \frac{13}{35} a^{8} - \frac{13}{35} a^{7} + \frac{1}{35} a^{6} + \frac{13}{35} a^{5} - \frac{1}{35} a^{4} + \frac{6}{35} a^{3} - \frac{3}{7} a^{2} + \frac{6}{35} a - \frac{3}{7}$, $\frac{1}{35} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{9}{35} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{3}{7}$, $\frac{1}{35} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{7} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{13}{35} a$, $\frac{1}{35} a^{14} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{9}{35} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{3} - \frac{13}{35} a^{2} - \frac{1}{5} a$, $\frac{1}{35} a^{15} - \frac{2}{5} a^{10} + \frac{1}{7} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{6}{35} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{35} a^{16} - \frac{2}{35} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{5} + \frac{3}{7} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{175} a^{17} - \frac{1}{175} a^{16} + \frac{1}{175} a^{15} + \frac{1}{175} a^{13} + \frac{1}{175} a^{12} - \frac{2}{175} a^{11} - \frac{54}{175} a^{10} + \frac{61}{175} a^{9} + \frac{1}{5} a^{8} - \frac{2}{175} a^{7} - \frac{16}{175} a^{6} + \frac{57}{175} a^{5} + \frac{4}{35} a^{4} - \frac{11}{35} a^{3} - \frac{3}{25} a^{2} - \frac{62}{175} a + \frac{2}{7}$, $\frac{1}{175} a^{18} + \frac{1}{175} a^{15} + \frac{1}{175} a^{14} + \frac{2}{175} a^{13} - \frac{1}{175} a^{12} - \frac{1}{175} a^{11} - \frac{8}{175} a^{10} + \frac{81}{175} a^{9} + \frac{18}{175} a^{8} - \frac{33}{175} a^{7} - \frac{79}{175} a^{6} - \frac{83}{175} a^{5} + \frac{17}{35} a^{4} + \frac{79}{175} a^{3} - \frac{33}{175} a^{2} - \frac{32}{175} a - \frac{3}{7}$, $\frac{1}{1225} a^{19} + \frac{1}{1225} a^{18} + \frac{3}{1225} a^{17} - \frac{1}{175} a^{16} - \frac{3}{245} a^{15} + \frac{8}{1225} a^{14} - \frac{6}{1225} a^{13} + \frac{16}{1225} a^{12} + \frac{3}{245} a^{11} - \frac{82}{175} a^{10} + \frac{37}{1225} a^{9} - \frac{41}{245} a^{8} + \frac{457}{1225} a^{7} + \frac{3}{35} a^{6} - \frac{32}{1225} a^{5} - \frac{18}{175} a^{4} + \frac{216}{1225} a^{3} - \frac{328}{1225} a^{2} - \frac{333}{1225} a + \frac{15}{49}$, $\frac{1}{1225} a^{20} + \frac{2}{1225} a^{18} - \frac{3}{1225} a^{17} - \frac{3}{245} a^{16} - \frac{1}{245} a^{15} - \frac{2}{175} a^{14} - \frac{6}{1225} a^{13} + \frac{6}{1225} a^{12} - \frac{8}{1225} a^{11} + \frac{93}{1225} a^{10} + \frac{121}{245} a^{9} - \frac{213}{1225} a^{8} - \frac{436}{1225} a^{7} + \frac{101}{1225} a^{6} - \frac{107}{245} a^{5} - \frac{113}{1225} a^{4} - \frac{579}{1225} a^{3} - \frac{12}{1225} a^{2} + \frac{379}{1225} a - \frac{15}{49}$, $\frac{1}{1225} a^{21} + \frac{2}{1225} a^{18} - \frac{12}{1225} a^{16} + \frac{9}{1225} a^{15} - \frac{3}{245} a^{14} - \frac{17}{1225} a^{13} + \frac{9}{1225} a^{12} + \frac{2}{175} a^{11} + \frac{318}{1225} a^{10} - \frac{82}{175} a^{9} + \frac{69}{245} a^{8} + \frac{524}{1225} a^{7} + \frac{256}{1225} a^{6} - \frac{24}{175} a^{5} - \frac{292}{1225} a^{4} - \frac{591}{1225} a^{3} + \frac{608}{1225} a^{2} + \frac{82}{245} a + \frac{12}{49}$, $\frac{1}{1225} a^{22} - \frac{2}{1225} a^{18} + \frac{3}{1225} a^{17} + \frac{2}{1225} a^{16} + \frac{1}{1225} a^{15} + \frac{2}{1225} a^{14} + \frac{1}{175} a^{13} + \frac{3}{1225} a^{12} + \frac{1}{1225} a^{11} + \frac{1}{7} a^{10} - \frac{338}{1225} a^{9} + \frac{129}{1225} a^{8} - \frac{4}{35} a^{7} + \frac{38}{175} a^{6} + \frac{234}{1225} a^{5} + \frac{326}{1225} a^{4} - \frac{34}{1225} a^{3} - \frac{113}{245} a^{2} + \frac{87}{175} a + \frac{12}{49}$, $\frac{1}{44032625} a^{23} - \frac{16643}{44032625} a^{22} - \frac{1244}{8806525} a^{21} + \frac{1332}{44032625} a^{20} + \frac{916}{44032625} a^{19} + \frac{65157}{44032625} a^{18} - \frac{55838}{44032625} a^{17} - \frac{430902}{44032625} a^{16} + \frac{10729}{898625} a^{15} + \frac{143903}{44032625} a^{14} + \frac{116989}{8806525} a^{13} - \frac{512223}{44032625} a^{12} - \frac{415147}{44032625} a^{11} + \frac{16666339}{44032625} a^{10} - \frac{178393}{3387125} a^{9} + \frac{392046}{1258075} a^{8} + \frac{2794322}{8806525} a^{7} - \frac{2967036}{6290375} a^{6} + \frac{20112137}{44032625} a^{5} + \frac{19998639}{44032625} a^{4} + \frac{15494541}{44032625} a^{3} - \frac{5663993}{44032625} a^{2} - \frac{18964937}{44032625} a + \frac{146378}{1761305}$, $\frac{1}{1893402875} a^{24} + \frac{8}{1893402875} a^{23} - \frac{24336}{145646375} a^{22} - \frac{441683}{1893402875} a^{21} - \frac{321522}{1893402875} a^{20} - \frac{462382}{1893402875} a^{19} + \frac{2892924}{1893402875} a^{18} + \frac{178098}{75736115} a^{17} + \frac{7531299}{1893402875} a^{16} + \frac{17276354}{1893402875} a^{15} + \frac{18880158}{1893402875} a^{14} + \frac{6759047}{1893402875} a^{13} - \frac{3906692}{378680575} a^{12} - \frac{3374544}{270486125} a^{11} - \frac{3714890}{15147223} a^{10} - \frac{552542049}{1893402875} a^{9} + \frac{4312746}{75736115} a^{8} + \frac{701068378}{1893402875} a^{7} - \frac{223197}{677425} a^{6} - \frac{569685819}{1893402875} a^{5} - \frac{3650397}{54097225} a^{4} + \frac{789418808}{1893402875} a^{3} + \frac{7744369}{54097225} a^{2} + \frac{70928514}{270486125} a + \frac{29061023}{75736115}$, $\frac{1}{19310815922125} a^{25} - \frac{4408}{19310815922125} a^{24} - \frac{23323}{3862163184425} a^{23} + \frac{106054986}{2758687988875} a^{22} - \frac{726922324}{19310815922125} a^{21} + \frac{2721713062}{19310815922125} a^{20} - \frac{6880779358}{19310815922125} a^{19} + \frac{11722907778}{19310815922125} a^{18} - \frac{14857757319}{19310815922125} a^{17} - \frac{232017921272}{19310815922125} a^{16} - \frac{4892462241}{551737597775} a^{15} + \frac{1369907409}{1485447378625} a^{14} - \frac{240544771552}{19310815922125} a^{13} - \frac{66054933751}{19310815922125} a^{12} - \frac{128836764934}{19310815922125} a^{11} - \frac{133054197636}{297089475725} a^{10} + \frac{591948599141}{3862163184425} a^{9} - \frac{5240639872547}{19310815922125} a^{8} + \frac{1151008062952}{19310815922125} a^{7} + \frac{827061074447}{2758687988875} a^{6} - \frac{7599442030379}{19310815922125} a^{5} - \frac{1187173342723}{19310815922125} a^{4} + \frac{2134991397143}{19310815922125} a^{3} - \frac{201658403187}{551737597775} a^{2} + \frac{452859215253}{3862163184425} a + \frac{76389771697}{154486527377}$, $\frac{1}{6455856803373375125} a^{26} + \frac{22836}{922265257624767875} a^{25} - \frac{1021108381}{6455856803373375125} a^{24} + \frac{20333032619}{6455856803373375125} a^{23} + \frac{397398971682214}{6455856803373375125} a^{22} - \frac{322047617119439}{1291171360674675025} a^{21} - \frac{325656430370671}{6455856803373375125} a^{20} + \frac{26694702100179}{258234272134935005} a^{19} - \frac{12397429686334603}{6455856803373375125} a^{18} - \frac{280075592992714}{496604369490259625} a^{17} + \frac{85403018004263961}{6455856803373375125} a^{16} - \frac{273375427853587}{6455856803373375125} a^{15} - \frac{8032913868640408}{1291171360674675025} a^{14} - \frac{1136420283964997}{131752179660681125} a^{13} + \frac{58699501747074301}{6455856803373375125} a^{12} + \frac{942503246043277}{131752179660681125} a^{11} - \frac{255792108380049922}{1291171360674675025} a^{10} - \frac{365166064235056979}{922265257624767875} a^{9} - \frac{2463302219127411813}{6455856803373375125} a^{8} - \frac{1780238315162754739}{6455856803373375125} a^{7} + \frac{504155753554762306}{6455856803373375125} a^{6} + \frac{2306167105234353286}{6455856803373375125} a^{5} + \frac{985374921874962833}{6455856803373375125} a^{4} + \frac{692404113641684207}{6455856803373375125} a^{3} + \frac{13688155888648399}{1291171360674675025} a^{2} - \frac{1601871731447581313}{6455856803373375125} a + \frac{70612416027932867}{258234272134935005}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 16038112241729.484 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 16038112241729.484 \cdot 1}{2\sqrt{20191329826970487018697554420683199956085496127}}\approx 2.68478229796354$ (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

3.1.3647.1, 9.1.176906199770881.1

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 $27$ $27$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
521Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3647.2t1.a.a$1$ $ 7 \cdot 521 $ \(\Q(\sqrt{-3647}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3647.3t2.a.a$2$ $ 7 \cdot 521 $ 3.1.3647.1 $S_3$ (as 3T2) $1$ $0$
* 2.3647.9t3.a.c$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.9t3.a.a$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.9t3.a.b$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.27t8.a.e$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.f$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.a$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.h$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.c$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.i$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.g$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.d$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.b$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (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 *.