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Properties

Label 21.7.19503572777...7007.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,31^{7}\cdot 577^{9}$
Root discriminant $47.92$
Ramified primes $31, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -10, 21, -147, 241, -391, 1528, 1031, 2628, 1703, -2156, 412, -633, -571, 227, -188, 115, -20, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 18*x^19 - 20*x^18 + 115*x^17 - 188*x^16 + 227*x^15 - 571*x^14 - 633*x^13 + 412*x^12 - 2156*x^11 + 1703*x^10 + 2628*x^9 + 1031*x^8 + 1528*x^7 - 391*x^6 + 241*x^5 - 147*x^4 + 21*x^3 - 10*x^2 - x + 1)
 
gp: K = bnfinit(x^21 + 18*x^19 - 20*x^18 + 115*x^17 - 188*x^16 + 227*x^15 - 571*x^14 - 633*x^13 + 412*x^12 - 2156*x^11 + 1703*x^10 + 2628*x^9 + 1031*x^8 + 1528*x^7 - 391*x^6 + 241*x^5 - 147*x^4 + 21*x^3 - 10*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{21} + 18 x^{19} - 20 x^{18} + 115 x^{17} - 188 x^{16} + 227 x^{15} - 571 x^{14} - 633 x^{13} + 412 x^{12} - 2156 x^{11} + 1703 x^{10} + 2628 x^{9} + 1031 x^{8} + 1528 x^{7} - 391 x^{6} + 241 x^{5} - 147 x^{4} + 21 x^{3} - 10 x^{2} - x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-195035727779935942827580963181007007=-\,31^{7}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.92$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $31, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} - \frac{4}{13} a^{16} - \frac{4}{13} a^{15} - \frac{1}{13} a^{14} - \frac{1}{13} a^{13} - \frac{4}{13} a^{12} + \frac{2}{13} a^{11} + \frac{5}{13} a^{10} - \frac{6}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{2}{13} a^{6} + \frac{6}{13} a^{5} + \frac{4}{13} a^{4} + \frac{4}{13} a^{3} - \frac{3}{13} a^{2} + \frac{4}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{18} + \frac{6}{13} a^{16} - \frac{4}{13} a^{15} - \frac{5}{13} a^{14} + \frac{5}{13} a^{13} - \frac{1}{13} a^{12} + \frac{1}{13} a^{10} + \frac{3}{13} a^{9} - \frac{2}{13} a^{8} - \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{6}{13} a^{4} + \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{19} - \frac{6}{13} a^{16} + \frac{6}{13} a^{15} - \frac{2}{13} a^{14} + \frac{5}{13} a^{13} - \frac{2}{13} a^{12} + \frac{2}{13} a^{11} - \frac{1}{13} a^{10} - \frac{5}{13} a^{9} - \frac{6}{13} a^{8} - \frac{5}{13} a^{7} + \frac{1}{13} a^{6} - \frac{3}{13} a^{5} + \frac{2}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{6993505344346859626359218017} a^{20} + \frac{137182520446580065795053692}{6993505344346859626359218017} a^{19} + \frac{143873680524778428132996021}{6993505344346859626359218017} a^{18} - \frac{65976415096104736088732843}{6993505344346859626359218017} a^{17} - \frac{623465287009131472821772131}{6993505344346859626359218017} a^{16} + \frac{2587936956970762276108774881}{6993505344346859626359218017} a^{15} + \frac{2907986684173610148152618681}{6993505344346859626359218017} a^{14} - \frac{1958816652299964298131806777}{6993505344346859626359218017} a^{13} + \frac{2370824542825948851153009871}{6993505344346859626359218017} a^{12} + \frac{604340084697773649066092197}{6993505344346859626359218017} a^{11} - \frac{2043325286638205418619586375}{6993505344346859626359218017} a^{10} - \frac{2391132424548179288157173997}{6993505344346859626359218017} a^{9} - \frac{694764223781227223845336607}{6993505344346859626359218017} a^{8} + \frac{1165111532251832673273379568}{6993505344346859626359218017} a^{7} + \frac{1571160197151309470418165485}{6993505344346859626359218017} a^{6} - \frac{209798520466933933144149834}{6993505344346859626359218017} a^{5} - \frac{1975833228176912706680466862}{6993505344346859626359218017} a^{4} - \frac{3000478901743828000782787563}{6993505344346859626359218017} a^{3} - \frac{1154255681635998731997238227}{6993505344346859626359218017} a^{2} + \frac{2873484715934329560938565309}{6993505344346859626359218017} a + \frac{752738833243247432492140884}{6993505344346859626359218017}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3294250109.26 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_7$ (as 21T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 15 conjugacy class representatives for $S_3\times D_7$
Character table for $S_3\times D_7$

Intermediate fields

3.1.31.1, 7.7.192100033.1

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 $21$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ $21$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$31$31.7.0.1$x^{7} - x + 18$$1$$7$$0$$C_7$$[\ ]^{7}$
31.14.7.2$x^{14} - 887503681 x^{2} + 495227053998$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
577Data not computed