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Properties

Label 27.1.536...079.1
Degree $27$
Signature $[1, 13]$
Discriminant $-5.368\times 10^{44}$
Root discriminant $45.36$
Ramified primes $31, 89$
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 - 5*x^26 + 4*x^25 - 53*x^24 + 12*x^23 - 178*x^22 + 281*x^21 - 251*x^20 + 1170*x^19 - 548*x^18 + 1924*x^17 - 3164*x^16 + 1182*x^15 - 5392*x^14 + 2732*x^13 - 5123*x^12 + 7843*x^11 + 5402*x^10 + 11272*x^9 - 9146*x^8 - 13685*x^7 - 8959*x^6 + 8451*x^5 + 8381*x^4 + 1087*x^3 - 2808*x^2 - 714*x - 59)
 
gp: K = bnfinit(x^27 - 5*x^26 + 4*x^25 - 53*x^24 + 12*x^23 - 178*x^22 + 281*x^21 - 251*x^20 + 1170*x^19 - 548*x^18 + 1924*x^17 - 3164*x^16 + 1182*x^15 - 5392*x^14 + 2732*x^13 - 5123*x^12 + 7843*x^11 + 5402*x^10 + 11272*x^9 - 9146*x^8 - 13685*x^7 - 8959*x^6 + 8451*x^5 + 8381*x^4 + 1087*x^3 - 2808*x^2 - 714*x - 59, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59, -714, -2808, 1087, 8381, 8451, -8959, -13685, -9146, 11272, 5402, 7843, -5123, 2732, -5392, 1182, -3164, 1924, -548, 1170, -251, 281, -178, 12, -53, 4, -5, 1]);
 

\(x^{27} - 5 x^{26} + 4 x^{25} - 53 x^{24} + 12 x^{23} - 178 x^{22} + 281 x^{21} - 251 x^{20} + 1170 x^{19} - 548 x^{18} + 1924 x^{17} - 3164 x^{16} + 1182 x^{15} - 5392 x^{14} + 2732 x^{13} - 5123 x^{12} + 7843 x^{11} + 5402 x^{10} + 11272 x^{9} - 9146 x^{8} - 13685 x^{7} - 8959 x^{6} + 8451 x^{5} + 8381 x^{4} + 1087 x^{3} - 2808 x^{2} - 714 x - 59\)  Toggle raw display

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:  \(-536750065653068684465204465961269314591399079\)\(\medspace = -\,31^{13}\cdot 89^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.36$
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:  $31, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} + \frac{4}{31} a^{12} - \frac{13}{31} a^{9} - \frac{10}{31} a^{6} + \frac{9}{31} a^{3} + \frac{15}{31}$, $\frac{1}{31} a^{16} + \frac{4}{31} a^{13} - \frac{13}{31} a^{10} - \frac{10}{31} a^{7} + \frac{9}{31} a^{4} + \frac{15}{31} a$, $\frac{1}{31} a^{17} + \frac{4}{31} a^{14} - \frac{13}{31} a^{11} - \frac{10}{31} a^{8} + \frac{9}{31} a^{5} + \frac{15}{31} a^{2}$, $\frac{1}{31} a^{18} + \frac{2}{31} a^{12} + \frac{11}{31} a^{9} - \frac{13}{31} a^{6} + \frac{10}{31} a^{3} + \frac{2}{31}$, $\frac{1}{217} a^{19} - \frac{1}{217} a^{18} + \frac{3}{217} a^{17} + \frac{3}{217} a^{16} + \frac{3}{217} a^{15} + \frac{15}{31} a^{14} - \frac{79}{217} a^{13} - \frac{3}{31} a^{12} + \frac{85}{217} a^{11} - \frac{4}{31} a^{10} - \frac{19}{217} a^{9} + \frac{32}{217} a^{8} - \frac{74}{217} a^{7} + \frac{107}{217} a^{6} - \frac{66}{217} a^{5} - \frac{87}{217} a^{4} + \frac{48}{217} a^{3} + \frac{2}{31} a^{2} - \frac{11}{31} a - \frac{19}{217}$, $\frac{1}{217} a^{20} + \frac{2}{217} a^{18} - \frac{1}{217} a^{17} - \frac{1}{217} a^{16} + \frac{3}{217} a^{15} - \frac{2}{217} a^{14} + \frac{89}{217} a^{13} + \frac{78}{217} a^{12} - \frac{69}{217} a^{11} + \frac{44}{217} a^{10} + \frac{76}{217} a^{9} + \frac{4}{31} a^{8} + \frac{103}{217} a^{7} + \frac{6}{217} a^{6} + \frac{1}{217} a^{5} - \frac{102}{217} a^{4} - \frac{15}{217} a^{3} + \frac{7}{31} a^{2} + \frac{16}{217} a - \frac{75}{217}$, $\frac{1}{217} a^{21} + \frac{1}{217} a^{18} - \frac{3}{217} a^{16} - \frac{1}{217} a^{15} - \frac{3}{7} a^{14} + \frac{19}{217} a^{13} + \frac{1}{217} a^{12} - \frac{85}{217} a^{10} - \frac{25}{217} a^{9} - \frac{1}{7} a^{8} - \frac{9}{31} a^{7} - \frac{66}{217} a^{6} + \frac{3}{7} a^{5} - \frac{58}{217} a^{4} + \frac{16}{217} a^{3} + \frac{3}{7} a^{2} + \frac{79}{217} a - \frac{74}{217}$, $\frac{1}{2387} a^{22} - \frac{2}{2387} a^{21} - \frac{3}{2387} a^{20} + \frac{5}{2387} a^{19} - \frac{12}{2387} a^{18} - \frac{23}{2387} a^{17} + \frac{20}{2387} a^{16} + \frac{24}{2387} a^{15} + \frac{274}{2387} a^{14} - \frac{13}{77} a^{13} + \frac{345}{2387} a^{12} + \frac{162}{341} a^{11} - \frac{967}{2387} a^{10} - \frac{439}{2387} a^{9} - \frac{1126}{2387} a^{8} - \frac{328}{2387} a^{7} - \frac{51}{2387} a^{6} - \frac{118}{341} a^{5} - \frac{127}{2387} a^{4} - \frac{864}{2387} a^{3} - \frac{46}{217} a^{2} - \frac{115}{341} a - \frac{410}{2387}$, $\frac{1}{40579} a^{23} - \frac{4}{40579} a^{22} - \frac{87}{40579} a^{21} - \frac{1}{3689} a^{20} + \frac{4}{3689} a^{19} + \frac{342}{40579} a^{18} - \frac{16}{3689} a^{17} + \frac{160}{40579} a^{16} - \frac{632}{40579} a^{15} + \frac{9972}{40579} a^{14} - \frac{2639}{5797} a^{13} - \frac{1206}{40579} a^{12} - \frac{16358}{40579} a^{11} + \frac{10163}{40579} a^{10} - \frac{4901}{40579} a^{9} + \frac{493}{2387} a^{8} + \frac{120}{527} a^{7} + \frac{8626}{40579} a^{6} - \frac{12808}{40579} a^{5} + \frac{17320}{40579} a^{4} - \frac{5774}{40579} a^{3} - \frac{19835}{40579} a^{2} + \frac{811}{5797} a + \frac{653}{5797}$, $\frac{1}{771001} a^{24} - \frac{4}{771001} a^{23} + \frac{134}{771001} a^{22} + \frac{1604}{771001} a^{21} + \frac{316}{771001} a^{20} - \frac{797}{771001} a^{19} + \frac{9888}{771001} a^{18} + \frac{8354}{771001} a^{17} - \frac{10050}{771001} a^{16} + \frac{11910}{771001} a^{15} + \frac{184014}{771001} a^{14} - \frac{34169}{771001} a^{13} + \frac{246139}{771001} a^{12} + \frac{8951}{70091} a^{11} + \frac{35151}{771001} a^{10} + \frac{51}{133} a^{9} + \frac{374876}{771001} a^{8} - \frac{48126}{110143} a^{7} + \frac{20438}{70091} a^{6} - \frac{1601}{771001} a^{5} - \frac{3028}{24871} a^{4} - \frac{251171}{771001} a^{3} - \frac{16669}{40579} a^{2} + \frac{9634}{24871} a + \frac{18210}{45353}$, $\frac{1}{137837245777} a^{25} - \frac{86907}{137837245777} a^{24} - \frac{144255}{19691035111} a^{23} - \frac{27498974}{137837245777} a^{22} - \frac{147248501}{137837245777} a^{21} - \frac{122126111}{137837245777} a^{20} + \frac{14022163}{137837245777} a^{19} + \frac{78172981}{19691035111} a^{18} + \frac{496248595}{137837245777} a^{17} - \frac{16920363}{7254591883} a^{16} + \frac{952567821}{137837245777} a^{15} + \frac{23816274232}{137837245777} a^{14} + \frac{32333378926}{137837245777} a^{13} - \frac{603231661}{1790094101} a^{12} - \frac{34528413812}{137837245777} a^{11} - \frac{3840864464}{8108073281} a^{10} - \frac{19883354167}{137837245777} a^{9} + \frac{33935570679}{137837245777} a^{8} - \frac{49308752431}{137837245777} a^{7} - \frac{7312102130}{19691035111} a^{6} - \frac{48823855746}{137837245777} a^{5} - \frac{5829773240}{12530658707} a^{4} + \frac{50597300151}{137837245777} a^{3} + \frac{14604794346}{137837245777} a^{2} - \frac{1049494557}{12530658707} a - \frac{4065376829}{137837245777}$, $\frac{1}{1585266163681277} a^{26} - \frac{3280}{1585266163681277} a^{25} + \frac{7051370}{13321564400683} a^{24} + \frac{16866953281}{1585266163681277} a^{23} + \frac{191510935218}{1585266163681277} a^{22} + \frac{263406651070}{144115105789207} a^{21} - \frac{2534350127877}{1585266163681277} a^{20} - \frac{404390307374}{226466594811611} a^{19} - \frac{10310109893558}{1585266163681277} a^{18} - \frac{175170528437}{29910682333609} a^{17} + \frac{17965909179615}{1585266163681277} a^{16} - \frac{2658016675610}{1585266163681277} a^{15} - \frac{389160668977601}{1585266163681277} a^{14} - \frac{299290271040283}{1585266163681277} a^{13} - \frac{10434033243831}{51137618183267} a^{12} - \frac{306126668099282}{1585266163681277} a^{11} - \frac{705478879913429}{1585266163681277} a^{10} + \frac{503770391038424}{1585266163681277} a^{9} + \frac{350565412266371}{1585266163681277} a^{8} + \frac{300916872985796}{1585266163681277} a^{7} + \frac{159828196069154}{1585266163681277} a^{6} - \frac{64432160461195}{1585266163681277} a^{5} + \frac{112282182918900}{226466594811611} a^{4} - \frac{64232845972748}{1585266163681277} a^{3} - \frac{516009400955086}{1585266163681277} a^{2} + \frac{786210705043347}{1585266163681277} a - \frac{757029289669797}{1585266163681277}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 438273503820.3111 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 438273503820.3111 \cdot 1}{2\sqrt{536750065653068684465204465961269314591399079}}\approx 0.449984393152585$ (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

3.1.2759.1, 9.1.57943777150561.1

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 $27$ $27$ $27$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ $27$ R $27$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2759.2t1.a.a$1$ $ 31 \cdot 89 $ \(\Q(\sqrt{-2759}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2759.3t2.a.a$2$ $ 31 \cdot 89 $ 3.1.2759.1 $S_3$ (as 3T2) $1$ $0$
* 2.2759.9t3.a.b$2$ $ 31 \cdot 89 $ 9.1.57943777150561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2759.9t3.a.c$2$ $ 31 \cdot 89 $ 9.1.57943777150561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2759.9t3.a.a$2$ $ 31 \cdot 89 $ 9.1.57943777150561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2759.27t8.a.d$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.h$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.b$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.c$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.g$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.f$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.e$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.a$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2759.27t8.a.i$2$ $ 31 \cdot 89 $ 27.1.536750065653068684465204465961269314591399079.1 $D_{27}$ (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 *.