Properties

Label 27.3.458...208.1
Degree $27$
Signature $[3, 12]$
Discriminant $4.587\times 10^{55}$
Root discriminant \(115.22\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{27}$ (as 27T2392)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 2)
 
gp: K = bnfinit(y^27 - 5*y - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 5*x - 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 5*x - 2)
 

\( x^{27} - 5x - 2 \) 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:  $[3, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(45866665066876460435700187354179628500729234048268894208\) \(\medspace = 2^{27}\cdot 7\cdot 12805208083\cdot 398913576491\cdot 9557048364382464605843591\) 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:  \(115.22\)
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:  not computed
Ramified primes:   \(2\), \(7\), \(12805208083\), \(398913576491\), \(9557048364382464605843591\) 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:  $\Q(\sqrt{68346\!\cdots\!60322}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:   $6a^{26}+a^{25}+8a^{24}+2a^{23}+9a^{22}+2a^{21}+9a^{20}+6a^{18}-5a^{17}+a^{16}-11a^{15}-4a^{14}-16a^{13}-7a^{12}-19a^{11}-8a^{10}-19a^{9}-4a^{8}-14a^{7}+2a^{6}-10a^{5}+8a^{4}-3a^{3}+18a^{2}+6a-1$, $38a^{26}-15a^{25}+8a^{24}+3a^{22}+3a^{21}+5a^{20}+3a^{19}+4a^{18}+5a^{17}+3a^{16}+a^{15}+2a^{14}-a^{13}-4a^{12}-4a^{11}-5a^{10}-10a^{9}-9a^{8}-8a^{7}-11a^{6}-11a^{5}-6a^{4}-8a^{3}-7a^{2}-187$, $7a^{26}-9a^{25}+11a^{24}-11a^{23}+11a^{22}-12a^{21}+12a^{20}-12a^{19}+14a^{18}-15a^{17}+14a^{16}-15a^{15}+16a^{14}-15a^{13}+17a^{12}-20a^{11}+19a^{10}-19a^{9}+22a^{8}-21a^{7}+21a^{6}-26a^{5}+26a^{4}-23a^{3}+26a^{2}-28a-9$, $a^{26}+2a^{25}-a^{24}+a^{23}+2a^{22}-3a^{21}+a^{20}-2a^{19}-a^{18}-3a^{17}+a^{16}-2a^{15}+2a^{14}+3a^{12}+3a^{11}-a^{10}+4a^{9}-3a^{7}-3a^{6}+3a^{5}-12a^{4}+3a^{3}+2a^{2}-3a-1$, $3a^{26}-20a^{25}-42a^{24}-54a^{23}-52a^{22}-41a^{21}-21a^{20}+11a^{19}+50a^{18}+78a^{17}+89a^{16}+77a^{15}+54a^{14}+14a^{13}-44a^{12}-104a^{11}-137a^{10}-140a^{9}-113a^{8}-62a^{7}+16a^{6}+116a^{5}+200a^{4}+227a^{3}+212a^{2}+151a+37$, $9a^{26}+a^{25}-4a^{24}+a^{23}+5a^{21}-7a^{20}-4a^{19}+11a^{18}-9a^{17}-3a^{16}+a^{15}-2a^{14}+7a^{13}-15a^{12}-2a^{11}+19a^{10}-17a^{9}-4a^{8}+10a^{7}+6a^{6}-3a^{5}-16a^{4}+32a^{3}+14a^{2}-40a-15$, $125a^{26}-49a^{25}+20a^{24}-9a^{23}+4a^{22}-a^{21}-a^{19}+3a^{18}-2a^{17}+a^{16}+a^{14}-4a^{13}+2a^{12}-2a^{10}+a^{9}+a^{8}-3a^{6}+5a^{5}-5a^{4}+3a-629$, $70a^{26}-3a^{25}-82a^{24}+19a^{23}+91a^{22}-38a^{21}-96a^{20}+60a^{19}+96a^{18}-84a^{17}-89a^{16}+109a^{15}+73a^{14}-134a^{13}-47a^{12}+157a^{11}+10a^{10}-176a^{9}+38a^{8}+188a^{7}-98a^{6}-192a^{5}+169a^{4}+186a^{3}-247a^{2}-164a-19$, $7a^{26}+4a^{25}+3a^{24}-10a^{23}+4a^{22}-6a^{21}+26a^{20}-41a^{19}+26a^{18}+a^{17}-6a^{16}-10a^{15}+15a^{14}-3a^{13}+16a^{12}-55a^{11}+62a^{10}-16a^{9}-16a^{8}+24a^{6}-13a^{5}+6a^{4}-48a^{3}+98a^{2}-74a-47$, $2a^{26}+2a^{25}-2a^{24}+2a^{23}+5a^{21}+a^{20}+4a^{19}-a^{18}+4a^{17}+5a^{15}+7a^{13}+3a^{12}+7a^{11}-3a^{10}+3a^{9}+a^{8}+14a^{7}+9a^{6}+12a^{5}-4a^{4}+2a^{3}+2a^{2}+23a+5$, $7a^{26}-9a^{25}+16a^{24}-16a^{23}+8a^{22}-5a^{21}-7a^{20}+22a^{19}-22a^{18}+15a^{17}-14a^{16}+3a^{15}+12a^{14}-11a^{13}+16a^{12}-31a^{11}+28a^{10}-10a^{9}+7a^{8}+18a^{7}-46a^{6}+39a^{5}-23a^{4}+24a^{3}-29a-15$, $2a^{26}+a^{25}-a^{24}+2a^{23}-2a^{21}+a^{20}-2a^{18}+5a^{17}-3a^{16}+4a^{15}-a^{14}+a^{13}-2a^{12}-a^{11}-a^{9}+4a^{8}+2a^{7}+2a^{4}-10a^{3}+10a^{2}-9a-7$, $13a^{26}-13a^{25}-a^{24}-a^{23}+5a^{22}+5a^{21}-16a^{20}+12a^{19}+2a^{18}+12a^{17}-10a^{16}-9a^{15}+14a^{14}-3a^{13}+a^{12}-28a^{11}+7a^{10}+17a^{9}-9a^{8}-17a^{6}+41a^{5}+9a^{4}-14a^{3}-13a^{2}+2a-29$, $9a^{26}+4a^{25}+4a^{24}-a^{23}-5a^{22}+13a^{21}+13a^{20}-14a^{19}-13a^{18}-2a^{17}-4a^{16}+13a^{15}+13a^{14}-17a^{13}-5a^{12}+27a^{11}+24a^{10}+13a^{9}-4a^{8}-23a^{7}-5a^{6}+21a^{5}+11a^{4}-27a^{3}-45a^{2}+4a+7$ 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:  \( 1078482016789482100 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 1078482016789482100 \cdot 1}{2\cdot\sqrt{45866665066876460435700187354179628500729234048268894208}}\cr\approx \mathstrut & 2.41147485303619 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 5*x - 2)
 
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 - 5*x - 2, 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 - 5*x - 2);
 
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 - 5*x - 2);
 
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

$S_{27}$ (as 27T2392):

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 non-solvable group of order 10888869450418352160768000000
The 3010 conjugacy class representatives for $S_{27}$ are not computed
Character table for $S_{27}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 R ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R $25{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $23{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ $19{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ $27$ ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$[3]$
Deg $24$$2$$12$$24$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.5.0.1$x^{5} + x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.7.0.1$x^{7} + 6 x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
7.12.0.1$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(12805208083\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(398913576491\) Copy content Toggle raw display $\Q_{398913576491}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$
\(955\!\cdots\!591\) Copy content Toggle raw display $\Q_{95\!\cdots\!91}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{95\!\cdots\!91}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{95\!\cdots\!91}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$