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Properties

Label 21.21.5553311747...0521.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 14197^{6}$
Root discriminant $56.20$
Ramified primes $7, 14197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -35, 159, 297, -1956, 422, 7595, -6566, -11778, 15158, 7395, -14826, -670, 6907, -1092, -1470, 386, 116, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 36*x^19 + 116*x^18 + 386*x^17 - 1470*x^16 - 1092*x^15 + 6907*x^14 - 670*x^13 - 14826*x^12 + 7395*x^11 + 15158*x^10 - 11778*x^9 - 6566*x^8 + 7595*x^7 + 422*x^6 - 1956*x^5 + 297*x^4 + 159*x^3 - 35*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^21 - 3*x^20 - 36*x^19 + 116*x^18 + 386*x^17 - 1470*x^16 - 1092*x^15 + 6907*x^14 - 670*x^13 - 14826*x^12 + 7395*x^11 + 15158*x^10 - 11778*x^9 - 6566*x^8 + 7595*x^7 + 422*x^6 - 1956*x^5 + 297*x^4 + 159*x^3 - 35*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 36 x^{19} + 116 x^{18} + 386 x^{17} - 1470 x^{16} - 1092 x^{15} + 6907 x^{14} - 670 x^{13} - 14826 x^{12} + 7395 x^{11} + 15158 x^{10} - 11778 x^{9} - 6566 x^{8} + 7595 x^{7} + 422 x^{6} - 1956 x^{5} + 297 x^{4} + 159 x^{3} - 35 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5553311747011057717852454287020950521=7^{14}\cdot 14197^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.20$
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:  $7, 14197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{1202302652530546601} a^{20} + \frac{284194788618636522}{1202302652530546601} a^{19} - \frac{272231351297699669}{1202302652530546601} a^{18} + \frac{278628161779132041}{1202302652530546601} a^{17} + \frac{757816131400755}{1202302652530546601} a^{16} - \frac{591535517204339106}{1202302652530546601} a^{15} + \frac{470485415669953990}{1202302652530546601} a^{14} - \frac{248262204643810589}{1202302652530546601} a^{13} - \frac{29611604231535447}{1202302652530546601} a^{12} - \frac{471525344389721235}{1202302652530546601} a^{11} + \frac{506225686189333484}{1202302652530546601} a^{10} - \frac{127499040595073214}{1202302652530546601} a^{9} - \frac{369796432047233084}{1202302652530546601} a^{8} - \frac{254073910867147079}{1202302652530546601} a^{7} - \frac{224267611614760987}{1202302652530546601} a^{6} - \frac{401149129937945043}{1202302652530546601} a^{5} + \frac{132040482403077655}{1202302652530546601} a^{4} + \frac{31835501679983929}{1202302652530546601} a^{3} + \frac{263675811508153032}{1202302652530546601} a^{2} + \frac{109382978881323253}{1202302652530546601} a + \frac{241019747553482355}{1202302652530546601}$

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:  \( 348948085655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1029
The 133 conjugacy class representatives for t21n28 are not computed
Character table for t21n28 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 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 $21$ $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ $21$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ $21$

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
7Data not computed
14197Data not computed