Properties

Label 21.9.252...441.1
Degree $21$
Signature $[9, 6]$
Discriminant $2.524\times 10^{27}$
Root discriminant $20.18$
Ramified primes $3, 73$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T44

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3)
 
gp: K = bnfinit(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -3, -8, 18, -51, 45, 54, -177, 63, 0, -21, 198, -90, -75, 66, 0, 3, 0, -9, 0, 1]);
 

\(x^{21} - 9 x^{19} + 3 x^{17} + 66 x^{15} - 75 x^{14} - 90 x^{13} + 198 x^{12} - 21 x^{11} + 63 x^{9} - 177 x^{8} + 54 x^{7} + 45 x^{6} - 51 x^{5} + 18 x^{4} - 8 x^{3} - 3 x^{2} + 3\)  Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2523828389200110188894232441\)\(\medspace = 3^{34}\cdot 73^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.18$
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:  $3, 73$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} + \frac{6}{29} a^{17} - \frac{2}{29} a^{16} - \frac{7}{29} a^{15} - \frac{9}{29} a^{14} - \frac{6}{29} a^{13} - \frac{5}{29} a^{12} - \frac{5}{29} a^{11} - \frac{5}{29} a^{10} - \frac{2}{29} a^{9} + \frac{4}{29} a^{7} - \frac{10}{29} a^{6} - \frac{5}{29} a^{5} - \frac{14}{29} a^{4} - \frac{2}{29} a^{3} - \frac{1}{29} a^{2} + \frac{11}{29}$, $\frac{1}{29} a^{19} - \frac{9}{29} a^{17} + \frac{5}{29} a^{16} + \frac{4}{29} a^{15} - \frac{10}{29} a^{14} + \frac{2}{29} a^{13} - \frac{4}{29} a^{12} - \frac{4}{29} a^{11} - \frac{1}{29} a^{10} + \frac{12}{29} a^{9} + \frac{4}{29} a^{8} - \frac{5}{29} a^{7} - \frac{3}{29} a^{6} - \frac{13}{29} a^{5} - \frac{5}{29} a^{4} + \frac{11}{29} a^{3} + \frac{6}{29} a^{2} + \frac{11}{29} a - \frac{8}{29}$, $\frac{1}{554583883475930137031} a^{20} + \frac{1543879837230387866}{554583883475930137031} a^{19} - \frac{9338021171565003108}{554583883475930137031} a^{18} - \frac{38778221679230729329}{554583883475930137031} a^{17} - \frac{99023689472789185765}{554583883475930137031} a^{16} + \frac{74981095580659074871}{554583883475930137031} a^{15} + \frac{21178924124033884797}{554583883475930137031} a^{14} + \frac{129051592554330205204}{554583883475930137031} a^{13} - \frac{208274738567600147827}{554583883475930137031} a^{12} + \frac{248192584477477154597}{554583883475930137031} a^{11} + \frac{169803346736758291664}{554583883475930137031} a^{10} - \frac{75990319030138058373}{554583883475930137031} a^{9} + \frac{245608091654983893030}{554583883475930137031} a^{8} + \frac{206977738496375222384}{554583883475930137031} a^{7} - \frac{217488545806445506167}{554583883475930137031} a^{6} + \frac{216642111496161310247}{554583883475930137031} a^{5} + \frac{106946772051958925848}{554583883475930137031} a^{4} - \frac{214875708717009555993}{554583883475930137031} a^{3} + \frac{172545894997721847072}{554583883475930137031} a^{2} + \frac{34515952381658016192}{554583883475930137031} a + \frac{104690597026483562860}{554583883475930137031}$  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:  $14$
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:  \( 823613.327719 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{6}\cdot 823613.327719 \cdot 1}{2\sqrt{2523828389200110188894232441}}\approx 0.258233529183$ (assuming GRH)

Galois group

21T44:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 7560
The 27 conjugacy class representatives for t21n44
Character table for t21n44 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.3.3884841.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 45 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $21$ R $21$ $21$ $21$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$