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Properties

Label 21.21.3406163121...4656.2
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{38}$
Root discriminant $61.27$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7\times C_7:C_3$ (as 21T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -10752, -100352, -164864, 200704, 524160, -150528, -658368, 53760, 448448, -9856, -183456, 896, 47040, -32, -7616, 0, 756, 0, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 32*x^14 + 47040*x^13 + 896*x^12 - 183456*x^11 - 9856*x^10 + 448448*x^9 + 53760*x^8 - 658368*x^7 - 150528*x^6 + 524160*x^5 + 200704*x^4 - 164864*x^3 - 100352*x^2 - 10752*x + 512)
 
gp: K = bnfinit(x^21 - 42*x^19 + 756*x^17 - 7616*x^15 - 32*x^14 + 47040*x^13 + 896*x^12 - 183456*x^11 - 9856*x^10 + 448448*x^9 + 53760*x^8 - 658368*x^7 - 150528*x^6 + 524160*x^5 + 200704*x^4 - 164864*x^3 - 100352*x^2 - 10752*x + 512, 1)
 

Normalized defining polynomial

\( x^{21} - 42 x^{19} + 756 x^{17} - 7616 x^{15} - 32 x^{14} + 47040 x^{13} + 896 x^{12} - 183456 x^{11} - 9856 x^{10} + 448448 x^{9} + 53760 x^{8} - 658368 x^{7} - 150528 x^{6} + 524160 x^{5} + 200704 x^{4} - 164864 x^{3} - 100352 x^{2} - 10752 x + 512 \)

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:  \(34061631211994616985595974961974214656=2^{18}\cdot 7^{38}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.27$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{8} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{16} a^{10}$, $\frac{1}{16} a^{11}$, $\frac{1}{32} a^{12}$, $\frac{1}{32} a^{13}$, $\frac{1}{64} a^{14}$, $\frac{1}{64} a^{15}$, $\frac{1}{64} a^{16}$, $\frac{1}{128} a^{17}$, $\frac{1}{128} a^{18}$, $\frac{1}{256} a^{19}$, $\frac{1}{256} a^{20}$

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:  \( 1354633483050 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times C_7:C_3$ (as 21T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147
The 35 conjugacy class representatives for $C_7\times C_7:C_3$
Character table for $C_7\times C_7:C_3$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 21 sibling: 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 $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ $21$ $21$ $21$

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
2Data not computed
7Data not computed