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Properties

Label 21.21.2243640041...3392.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{20}\cdot 7^{30}\cdot 37^{7}$
Root discriminant $103.92$
Ramified primes $2, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times C_7:C_3$ (as 21T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 56, -392, -3136, 2352, 22596, -6468, -60656, 8456, 80164, -5096, -57624, 1274, 23324, -106, -5348, 0, 672, 0, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 42*x^19 + 672*x^17 - 5348*x^15 - 106*x^14 + 23324*x^13 + 1274*x^12 - 57624*x^11 - 5096*x^10 + 80164*x^9 + 8456*x^8 - 60656*x^7 - 6468*x^6 + 22596*x^5 + 2352*x^4 - 3136*x^3 - 392*x^2 + 56*x + 4)
 
gp: K = bnfinit(x^21 - 42*x^19 + 672*x^17 - 5348*x^15 - 106*x^14 + 23324*x^13 + 1274*x^12 - 57624*x^11 - 5096*x^10 + 80164*x^9 + 8456*x^8 - 60656*x^7 - 6468*x^6 + 22596*x^5 + 2352*x^4 - 3136*x^3 - 392*x^2 + 56*x + 4, 1)
 

Normalized defining polynomial

\( x^{21} - 42 x^{19} + 672 x^{17} - 5348 x^{15} - 106 x^{14} + 23324 x^{13} + 1274 x^{12} - 57624 x^{11} - 5096 x^{10} + 80164 x^{9} + 8456 x^{8} - 60656 x^{7} - 6468 x^{6} + 22596 x^{5} + 2352 x^{4} - 3136 x^{3} - 392 x^{2} + 56 x + 4 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2243640041810033585224532070313948733243392=2^{20}\cdot 7^{30}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.92$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{17} - \frac{1}{6} a^{16} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{19} + \frac{1}{6} a^{17} - \frac{1}{6} a^{16} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{2271172801722137845940274} a^{20} - \frac{62954045962662403916795}{757057600574045948646758} a^{19} + \frac{13025645081653118332519}{757057600574045948646758} a^{18} + \frac{224556514660597434558734}{1135586400861068922970137} a^{17} - \frac{29226452261292383381374}{1135586400861068922970137} a^{16} - \frac{149417236232686703315192}{1135586400861068922970137} a^{15} - \frac{213617021569683919451660}{1135586400861068922970137} a^{14} + \frac{12471554576566200650660}{378528800287022974323379} a^{13} - \frac{118118038966198251593399}{757057600574045948646758} a^{12} + \frac{17868785346901872546179}{757057600574045948646758} a^{11} + \frac{505554774441171485748748}{1135586400861068922970137} a^{10} - \frac{320928215508665481446842}{1135586400861068922970137} a^{9} + \frac{169930758295148334374114}{378528800287022974323379} a^{8} - \frac{172046780603254801360554}{378528800287022974323379} a^{7} - \frac{567192317100268217982332}{1135586400861068922970137} a^{6} + \frac{410177444805919198019953}{1135586400861068922970137} a^{5} - \frac{36865588142204094992963}{378528800287022974323379} a^{4} - \frac{181491978390946141225006}{1135586400861068922970137} a^{3} + \frac{277532795190916028417110}{1135586400861068922970137} a^{2} - \frac{381255193446112938194708}{1135586400861068922970137} a + \frac{170148610685133761487944}{1135586400861068922970137}$

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:  $20$
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:  \( 798369236455000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times C_7:C_3$ (as 21T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 15 conjugacy class representatives for $S_3\times C_7:C_3$
Character table for $S_3\times C_7:C_3$

Intermediate fields

3.3.148.1, 7.7.18078415936.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $21$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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
2Data not computed
7Data not computed
37Data not computed