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Properties

Label 21.9.11159702690...6281.2
Degree $21$
Signature $[9, 6]$
Discriminant $13^{16}\cdot 109^{6}$
Root discriminant $26.97$
Ramified primes $13, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times \PSL(2,7)$ (as 21T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79, 1631, -1996, -5649, 9936, 3035, -14456, 4203, 6562, -4414, 1019, 1150, -1078, -7, 123, -94, -14, 49, 8, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 12*x^19 + 8*x^18 + 49*x^17 - 14*x^16 - 94*x^15 + 123*x^14 - 7*x^13 - 1078*x^12 + 1150*x^11 + 1019*x^10 - 4414*x^9 + 6562*x^8 + 4203*x^7 - 14456*x^6 + 3035*x^5 + 9936*x^4 - 5649*x^3 - 1996*x^2 + 1631*x - 79)
 
gp: K = bnfinit(x^21 - x^20 - 12*x^19 + 8*x^18 + 49*x^17 - 14*x^16 - 94*x^15 + 123*x^14 - 7*x^13 - 1078*x^12 + 1150*x^11 + 1019*x^10 - 4414*x^9 + 6562*x^8 + 4203*x^7 - 14456*x^6 + 3035*x^5 + 9936*x^4 - 5649*x^3 - 1996*x^2 + 1631*x - 79, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 12 x^{19} + 8 x^{18} + 49 x^{17} - 14 x^{16} - 94 x^{15} + 123 x^{14} - 7 x^{13} - 1078 x^{12} + 1150 x^{11} + 1019 x^{10} - 4414 x^{9} + 6562 x^{8} + 4203 x^{7} - 14456 x^{6} + 3035 x^{5} + 9936 x^{4} - 5649 x^{3} - 1996 x^{2} + 1631 x - 79 \)

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:  \(1115970269016553289813936756281=13^{16}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
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:  $13, 109$
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}{561034684070749862481847410909310375152731} a^{20} - \frac{237453031596959799903235018619163827713867}{561034684070749862481847410909310375152731} a^{19} - \frac{58068034783322943716590538695717706033787}{561034684070749862481847410909310375152731} a^{18} - \frac{121779857785771951366576783422434476213729}{561034684070749862481847410909310375152731} a^{17} + \frac{99941762291471144391801274591165543871387}{561034684070749862481847410909310375152731} a^{16} - \frac{4427474690987737058268689603186600648090}{43156514159288450960142108531485413473287} a^{15} - \frac{32474667626737125547576226389429889713639}{561034684070749862481847410909310375152731} a^{14} - \frac{206209840435817698991827653886593418049951}{561034684070749862481847410909310375152731} a^{13} + \frac{140464067057230742869247160574661376835208}{561034684070749862481847410909310375152731} a^{12} + \frac{121111374693046544604316009044949910653031}{561034684070749862481847410909310375152731} a^{11} - \frac{148761341497779463853151431231130594915889}{561034684070749862481847410909310375152731} a^{10} + \frac{85304839928811439212215518421700954052636}{561034684070749862481847410909310375152731} a^{9} - \frac{104212917291410479322049371420246980211818}{561034684070749862481847410909310375152731} a^{8} - \frac{424592141601804932441790975776204314729}{561034684070749862481847410909310375152731} a^{7} + \frac{66486652213222665175011667840241969387433}{561034684070749862481847410909310375152731} a^{6} - \frac{48569627133344085361225689105779739718918}{561034684070749862481847410909310375152731} a^{5} - \frac{212994990657287635554080598122978358620432}{561034684070749862481847410909310375152731} a^{4} - \frac{142838332387389987685664468307556515912621}{561034684070749862481847410909310375152731} a^{3} - \frac{111625143178480064936161787665248779052898}{561034684070749862481847410909310375152731} a^{2} + \frac{18048977935636737412591779347981172062741}{561034684070749862481847410909310375152731} a + \frac{7308429899422233610569251835537483130491}{561034684070749862481847410909310375152731}$

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:  \( 12045126.3727 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times \PSL(2,7)$ (as 21T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 504
The 18 conjugacy class representatives for $C_3\times \PSL(2,7)$
Character table for $C_3\times \PSL(2,7)$

Intermediate fields

3.3.169.1, 7.3.2007889.1

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

Sibling fields

Degree 24 siblings: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently 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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.5$x^{12} + 65 x^{6} + 1352$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
109Data not computed