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Properties

Label 21.21.3543390101...4944.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{22}\cdot 4129^{8}$
Root discriminant $49.30$
Ramified primes $2, 4129$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,7)$ (as 21T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -8, -756, 6574, -14068, -26074, 106634, -29458, -165588, 98768, 110250, -79894, -39018, 30708, 8040, -6395, -988, 733, 68, -43, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 43*x^19 + 68*x^18 + 733*x^17 - 988*x^16 - 6395*x^15 + 8040*x^14 + 30708*x^13 - 39018*x^12 - 79894*x^11 + 110250*x^10 + 98768*x^9 - 165588*x^8 - 29458*x^7 + 106634*x^6 - 26074*x^5 - 14068*x^4 + 6574*x^3 - 756*x^2 - 8*x + 4)
 
gp: K = bnfinit(x^21 - 2*x^20 - 43*x^19 + 68*x^18 + 733*x^17 - 988*x^16 - 6395*x^15 + 8040*x^14 + 30708*x^13 - 39018*x^12 - 79894*x^11 + 110250*x^10 + 98768*x^9 - 165588*x^8 - 29458*x^7 + 106634*x^6 - 26074*x^5 - 14068*x^4 + 6574*x^3 - 756*x^2 - 8*x + 4, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 43 x^{19} + 68 x^{18} + 733 x^{17} - 988 x^{16} - 6395 x^{15} + 8040 x^{14} + 30708 x^{13} - 39018 x^{12} - 79894 x^{11} + 110250 x^{10} + 98768 x^{9} - 165588 x^{8} - 29458 x^{7} + 106634 x^{6} - 26074 x^{5} - 14068 x^{4} + 6574 x^{3} - 756 x^{2} - 8 x + 4 \)

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:  \(354339010128398236785800917962194944=2^{22}\cdot 4129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.30$
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:  $2, 4129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{112} a^{18} + \frac{3}{112} a^{17} - \frac{23}{112} a^{16} - \frac{17}{112} a^{15} + \frac{9}{56} a^{14} + \frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{8} a^{11} + \frac{9}{56} a^{10} - \frac{17}{56} a^{9} + \frac{9}{28} a^{8} - \frac{1}{8} a^{7} + \frac{2}{7} a^{6} + \frac{9}{56} a^{5} + \frac{19}{56} a^{4} - \frac{1}{8} a^{3} - \frac{9}{28} a^{2} + \frac{1}{4} a + \frac{13}{28}$, $\frac{1}{448} a^{19} - \frac{1}{112} a^{17} - \frac{1}{112} a^{16} - \frac{71}{448} a^{15} - \frac{19}{224} a^{14} - \frac{11}{56} a^{13} - \frac{73}{224} a^{12} - \frac{3}{56} a^{11} + \frac{17}{56} a^{10} + \frac{41}{224} a^{9} - \frac{61}{224} a^{8} - \frac{19}{224} a^{7} + \frac{101}{224} a^{6} - \frac{9}{56} a^{5} - \frac{2}{7} a^{4} + \frac{87}{224} a^{3} - \frac{11}{56} a^{2} + \frac{3}{7} a - \frac{39}{112}$, $\frac{1}{64908932073646612411264} a^{20} + \frac{19020507732273282493}{64908932073646612411264} a^{19} - \frac{72232982023965513483}{16227233018411653102816} a^{18} - \frac{96404187614170165453}{4056808254602913275704} a^{17} + \frac{145084444610714208059}{9272704581949516058752} a^{16} - \frac{10462198930791837384361}{64908932073646612411264} a^{15} + \frac{12418432170456579637}{32454466036823306205632} a^{14} + \frac{5372066373059324021675}{32454466036823306205632} a^{13} + \frac{12320939786678765230271}{32454466036823306205632} a^{12} - \frac{232454815407912228833}{2028404127301456637852} a^{11} - \frac{863543984308370411371}{32454466036823306205632} a^{10} + \frac{308343456616231060179}{2028404127301456637852} a^{9} + \frac{1749086485154982716349}{8113616509205826551408} a^{8} + \frac{2404864391735032720651}{16227233018411653102816} a^{7} - \frac{14879366014576317284563}{32454466036823306205632} a^{6} - \frac{1910514100603095319543}{8113616509205826551408} a^{5} - \frac{5115703795239318650081}{32454466036823306205632} a^{4} - \frac{3009165760877217635545}{32454466036823306205632} a^{3} - \frac{3165741525405458943011}{8113616509205826551408} a^{2} + \frac{6932004577273342017441}{16227233018411653102816} a + \frac{1275528559471777447997}{16227233018411653102816}$

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:  \( 646551468945 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,7)$ (as 21T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

7.7.1091113024.1, 7.7.1091113024.2

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

Sibling fields

Degree 7 siblings: 7.7.1091113024.1, 7.7.1091113024.2
Degree 8 sibling: 8.8.74407976946401536.1
Degree 14 siblings: Deg 14, Deg 14
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.14.1$x^{12} + 2 x^{3} + 2$$12$$1$$14$$S_4$$[4/3, 4/3]_{3}^{2}$
4129Data not computed