↑

Properties

Label 21.9.15564055280...7113.1
Degree $21$
Signature $[9, 6]$
Discriminant $7^{14}\cdot 71^{3}\cdot 8623^{3}$
Root discriminant $24.55$
Ramified primes $7, 71, 8623$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T56

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -7, 9, 3, -71, 140, -244, -80, 362, 86, 84, -54, -99, -57, -38, 11, -8, 0, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 3*x^19 - 8*x^17 + 11*x^16 - 38*x^15 - 57*x^14 - 99*x^13 - 54*x^12 + 84*x^11 + 86*x^10 + 362*x^9 - 80*x^8 - 244*x^7 + 140*x^6 - 71*x^5 + 3*x^4 + 9*x^3 - 7*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^21 - x^20 + 3*x^19 - 8*x^17 + 11*x^16 - 38*x^15 - 57*x^14 - 99*x^13 - 54*x^12 + 84*x^11 + 86*x^10 + 362*x^9 - 80*x^8 - 244*x^7 + 140*x^6 - 71*x^5 + 3*x^4 + 9*x^3 - 7*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} + 3 x^{19} - 8 x^{17} + 11 x^{16} - 38 x^{15} - 57 x^{14} - 99 x^{13} - 54 x^{12} + 84 x^{11} + 86 x^{10} + 362 x^{9} - 80 x^{8} - 244 x^{7} + 140 x^{6} - 71 x^{5} + 3 x^{4} + 9 x^{3} - 7 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(155640552803090077912213347113=7^{14}\cdot 71^{3}\cdot 8623^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.55$
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:  $7, 71, 8623$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{3}{23} a^{17} - \frac{4}{23} a^{16} + \frac{11}{23} a^{15} - \frac{4}{23} a^{14} + \frac{2}{23} a^{13} + \frac{4}{23} a^{12} - \frac{1}{23} a^{11} - \frac{11}{23} a^{10} + \frac{4}{23} a^{9} - \frac{5}{23} a^{8} + \frac{9}{23} a^{7} + \frac{1}{23} a^{6} + \frac{10}{23} a^{4} + \frac{11}{23} a^{3} - \frac{10}{23} a^{2} - \frac{4}{23} a + \frac{3}{23}$, $\frac{1}{23} a^{19} + \frac{10}{23} a^{17} + \frac{9}{23} a^{15} - \frac{9}{23} a^{14} - \frac{2}{23} a^{13} + \frac{10}{23} a^{12} - \frac{8}{23} a^{11} - \frac{9}{23} a^{10} + \frac{6}{23} a^{9} + \frac{1}{23} a^{8} - \frac{3}{23} a^{7} - \frac{3}{23} a^{6} + \frac{10}{23} a^{5} + \frac{4}{23} a^{4} + \frac{3}{23} a^{3} + \frac{3}{23} a^{2} - \frac{8}{23} a - \frac{9}{23}$, $\frac{1}{4376424924938537193763} a^{20} + \frac{51652021723371314564}{4376424924938537193763} a^{19} - \frac{55907873299649760041}{4376424924938537193763} a^{18} - \frac{6362282514137072212}{190279344562545095381} a^{17} + \frac{1406312832693885503359}{4376424924938537193763} a^{16} - \frac{333617508304436477308}{4376424924938537193763} a^{15} + \frac{507655062887162933970}{4376424924938537193763} a^{14} - \frac{941265566122452520273}{4376424924938537193763} a^{13} + \frac{1957599841894456306586}{4376424924938537193763} a^{12} + \frac{1771843031667840950956}{4376424924938537193763} a^{11} - \frac{1141250932321788273186}{4376424924938537193763} a^{10} - \frac{1504602893921909686732}{4376424924938537193763} a^{9} + \frac{1712772511525964391775}{4376424924938537193763} a^{8} + \frac{1982257535881986594604}{4376424924938537193763} a^{7} - \frac{1821346203092740079174}{4376424924938537193763} a^{6} + \frac{641610731920765294363}{4376424924938537193763} a^{5} - \frac{443025497482896108851}{4376424924938537193763} a^{4} - \frac{402165530376285841847}{4376424924938537193763} a^{3} - \frac{1165883532532830877592}{4376424924938537193763} a^{2} + \frac{863186137706479669663}{4376424924938537193763} a - \frac{1993338329470358432300}{4376424924938537193763}$

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:  $14$
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:  \( 4976307.00235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T56:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15120
The 45 conjugacy class representatives for t21n56
Character table for t21n56 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.3.612233.1

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

Sibling fields

Degree 42 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 $21$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ $15{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ $15{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
71Data not computed
8623Data not computed