Properties

Label 21.1.990...104.1
Degree $21$
Signature $[1, 10]$
Discriminant $9.901\times 10^{27}$
Root discriminant $21.53$
Ramified primes $2, 11, 13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{21}:C_3$ (as 21T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38)
 
gp: K = bnfinit(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-38, 156, -142, -179, 669, -847, 387, 439, -1171, 1374, -907, 294, -57, 131, -215, 156, -24, -40, 25, -2, -3, 1]);
 

\( x^{21} - 3 x^{20} - 2 x^{19} + 25 x^{18} - 40 x^{17} - 24 x^{16} + 156 x^{15} - 215 x^{14} + 131 x^{13} - 57 x^{12} + 294 x^{11} - 907 x^{10} + 1374 x^{9} - 1171 x^{8} + 439 x^{7} + 387 x^{6} - 847 x^{5} + 669 x^{4} - 179 x^{3} - 142 x^{2} + 156 x - 38 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(9900725472657912423302447104\)\(\medspace = 2^{14}\cdot 11^{10}\cdot 13^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.53$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{39} a^{19} - \frac{2}{13} a^{17} - \frac{2}{13} a^{16} + \frac{5}{39} a^{15} - \frac{11}{39} a^{14} - \frac{1}{39} a^{13} - \frac{6}{13} a^{12} + \frac{1}{13} a^{11} + \frac{11}{39} a^{10} - \frac{10}{39} a^{9} - \frac{2}{13} a^{8} - \frac{8}{39} a^{7} - \frac{14}{39} a^{6} + \frac{1}{3} a^{5} + \frac{14}{39} a^{4} - \frac{4}{13} a^{3} - \frac{8}{39} a^{2} + \frac{14}{39} a - \frac{16}{39}$, $\frac{1}{5442110603004520264209} a^{20} - \frac{4762453487204400137}{1814036867668173421403} a^{19} + \frac{638787401185645951519}{5442110603004520264209} a^{18} - \frac{364237959278012756518}{5442110603004520264209} a^{17} - \frac{482498897405706721136}{5442110603004520264209} a^{16} + \frac{35476560329394141057}{1814036867668173421403} a^{15} + \frac{815886668501008054291}{1814036867668173421403} a^{14} - \frac{255618197087729759087}{1814036867668173421403} a^{13} + \frac{1046546015764076383040}{5442110603004520264209} a^{12} - \frac{138637131097717499896}{418623892538809251093} a^{11} + \frac{2633540085011196877691}{5442110603004520264209} a^{10} + \frac{2645087005385414060090}{5442110603004520264209} a^{9} - \frac{1637310832000467617395}{5442110603004520264209} a^{8} - \frac{1240883583523062976577}{5442110603004520264209} a^{7} + \frac{2514235604772958766045}{5442110603004520264209} a^{6} - \frac{2154369761976319252754}{5442110603004520264209} a^{5} - \frac{396827414586703850410}{1814036867668173421403} a^{4} + \frac{1960958511140047154551}{5442110603004520264209} a^{3} - \frac{169333224107209204462}{418623892538809251093} a^{2} - \frac{26538827258590311052}{5442110603004520264209} a - \frac{2122737236820434828284}{5442110603004520264209}$

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:  $10$
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:  \( 637026.16318 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{10}\cdot 637026.16318 \cdot 1}{2\sqrt{9900725472657912423302447104}}\approx 0.61393507198$ (assuming GRH)

Galois group

$D_{21}:C_3$ (as 21T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 126
The 12 conjugacy class representatives for $D_{21}:C_3$
Character table for $D_{21}:C_3$

Intermediate fields

3.1.44.1, 7.1.38014691.1

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

Sibling fields

Degree 42 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$