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Properties

Label 21.1.341...000.1
Degree $21$
Signature $[1, 10]$
Discriminant $3.415\times 10^{28}$
Root discriminant $22.84$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16)
 
gp: K = bnfinit(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 0, 0, -32, 0, 0, -144, 0, 0, -224, 0, 0, -76, 0, 0, 36, 0, 0, -10, 0, 0, 1]);
 

\( x^{21} - 10 x^{18} + 36 x^{15} - 76 x^{12} - 224 x^{9} - 144 x^{6} - 32 x^{3} - 16 \)

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:  \(34154868120917133312000000000\)\(\medspace = 2^{20}\cdot 3^{34}\cdot 5^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.84$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{18} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{36} a^{11} + \frac{1}{18} a^{9} + \frac{1}{9} a^{8} - \frac{1}{6} a^{7} + \frac{1}{18} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{36} a^{12} + \frac{1}{18} a^{9} - \frac{1}{6} a^{6} - \frac{2}{9} a^{3} + \frac{2}{9}$, $\frac{1}{36} a^{13} - \frac{1}{18} a^{9} + \frac{1}{6} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{36} a^{14} + \frac{1}{18} a^{9} + \frac{1}{9} a^{8} - \frac{1}{6} a^{7} + \frac{1}{18} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{108} a^{15} - \frac{1}{27} a^{9} - \frac{2}{27} a^{6} - \frac{1}{3} a^{3} + \frac{8}{27}$, $\frac{1}{648} a^{16} - \frac{1}{324} a^{15} - \frac{1}{108} a^{14} + \frac{1}{108} a^{12} + \frac{1}{108} a^{11} - \frac{1}{162} a^{10} - \frac{13}{162} a^{9} + \frac{1}{6} a^{8} - \frac{29}{162} a^{7} + \frac{2}{81} a^{6} - \frac{10}{27} a^{5} + \frac{4}{9} a^{4} - \frac{5}{27} a^{3} + \frac{13}{27} a^{2} - \frac{23}{81} a - \frac{20}{81}$, $\frac{1}{648} a^{17} + \frac{1}{324} a^{15} + \frac{1}{108} a^{14} + \frac{1}{108} a^{13} + \frac{1}{81} a^{11} + \frac{1}{54} a^{10} + \frac{7}{162} a^{9} - \frac{11}{162} a^{8} + \frac{1}{9} a^{7} + \frac{16}{81} a^{6} + \frac{10}{27} a^{5} - \frac{2}{27} a^{4} + \frac{37}{81} a^{2} - \frac{7}{27} a - \frac{10}{81}$, $\frac{1}{648} a^{18} - \frac{1}{324} a^{15} - \frac{1}{162} a^{12} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{31}{162} a^{6} + \frac{22}{81} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{26}{81}$, $\frac{1}{648} a^{19} + \frac{1}{324} a^{15} + \frac{1}{108} a^{14} - \frac{1}{162} a^{13} - \frac{1}{108} a^{12} - \frac{1}{108} a^{11} - \frac{1}{81} a^{10} + \frac{2}{81} a^{9} + \frac{1}{6} a^{8} - \frac{2}{9} a^{7} - \frac{13}{162} a^{6} + \frac{10}{27} a^{5} + \frac{4}{81} a^{4} + \frac{2}{27} a^{3} + \frac{5}{27} a^{2} - \frac{1}{3} a - \frac{16}{81}$, $\frac{1}{648} a^{20} - \frac{1}{324} a^{15} + \frac{1}{81} a^{14} - \frac{1}{108} a^{13} - \frac{1}{324} a^{11} - \frac{1}{54} a^{10} - \frac{7}{162} a^{9} + \frac{2}{9} a^{8} + \frac{1}{18} a^{7} + \frac{11}{81} a^{6} + \frac{28}{81} a^{5} + \frac{2}{27} a^{4} - \frac{2}{27} a^{2} - \frac{11}{27} a - \frac{17}{81}$

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:  \( 2085380.41806 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{10}\cdot 2085380.41806 \cdot 1}{2\sqrt{34154868120917133312000000000}}\approx 1.08207576000$ (assuming GRH)

Galois group

$S_3\times F_7$ (as 21T15):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.108.1, 7.1.157464000.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$