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Properties

Label 21.21.1918683763...7989.1
Degree $21$
Signature $[21, 0]$
Discriminant $107509^{9}$
Root discriminant $143.33$
Ramified prime $107509$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $SO(3,7)$ (as 21T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27082, -4643865, -20912877, 66372478, 109229003, -324673546, 116217006, 164165998, -100138483, -32803562, 29342562, 2641366, -4495452, 51461, 399755, -26782, -20809, 2119, 589, -74, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 74*x^19 + 589*x^18 + 2119*x^17 - 20809*x^16 - 26782*x^15 + 399755*x^14 + 51461*x^13 - 4495452*x^12 + 2641366*x^11 + 29342562*x^10 - 32803562*x^9 - 100138483*x^8 + 164165998*x^7 + 116217006*x^6 - 324673546*x^5 + 109229003*x^4 + 66372478*x^3 - 20912877*x^2 - 4643865*x + 27082)
 
gp: K = bnfinit(x^21 - 7*x^20 - 74*x^19 + 589*x^18 + 2119*x^17 - 20809*x^16 - 26782*x^15 + 399755*x^14 + 51461*x^13 - 4495452*x^12 + 2641366*x^11 + 29342562*x^10 - 32803562*x^9 - 100138483*x^8 + 164165998*x^7 + 116217006*x^6 - 324673546*x^5 + 109229003*x^4 + 66372478*x^3 - 20912877*x^2 - 4643865*x + 27082, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 74 x^{19} + 589 x^{18} + 2119 x^{17} - 20809 x^{16} - 26782 x^{15} + 399755 x^{14} + 51461 x^{13} - 4495452 x^{12} + 2641366 x^{11} + 29342562 x^{10} - 32803562 x^{9} - 100138483 x^{8} + 164165998 x^{7} + 116217006 x^{6} - 324673546 x^{5} + 109229003 x^{4} + 66372478 x^{3} - 20912877 x^{2} - 4643865 x + 27082 \)

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:  \(1918683763390753406969558352246443638814987989=107509^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.33$
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:  $107509$
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}{31006102317414334748740329077370791402380700735034781635575} a^{20} + \frac{2588600724815870726552184428014079044246596099669732672057}{31006102317414334748740329077370791402380700735034781635575} a^{19} - \frac{513120718246104963073243829487047900320184521936726024001}{31006102317414334748740329077370791402380700735034781635575} a^{18} - \frac{303859039189918961564162971823266726185352094043193495401}{1240244092696573389949613163094831656095228029401391265423} a^{17} - \frac{7375478157468636323454348991783014963604821334941731744656}{31006102317414334748740329077370791402380700735034781635575} a^{16} - \frac{108830825003135336760912018070870180821169652383528964926}{721072146916612436017216955287692823311179086861273991525} a^{15} + \frac{8266739675230202810547193958404004798111226400177858495566}{31006102317414334748740329077370791402380700735034781635575} a^{14} + \frac{10670056431931065624009172537714067112347951189428021740729}{31006102317414334748740329077370791402380700735034781635575} a^{13} - \frac{15259318198273799065167146429022020424545184146766160682808}{31006102317414334748740329077370791402380700735034781635575} a^{12} + \frac{14026295318061647077462446267011980820445052471371037733786}{31006102317414334748740329077370791402380700735034781635575} a^{11} + \frac{347066875633811421167831400838015661999306292422763529669}{6201220463482866949748065815474158280476140147006956327115} a^{10} - \frac{3814015958208666462802882596616312398928517453640178427158}{31006102317414334748740329077370791402380700735034781635575} a^{9} - \frac{3393650949821702876369213206788563085645293460101585241749}{31006102317414334748740329077370791402380700735034781635575} a^{8} + \frac{9125951048931441040715703561251010464954576188601105866856}{31006102317414334748740329077370791402380700735034781635575} a^{7} + \frac{10556467963018742867297348766816387061207116657137201276182}{31006102317414334748740329077370791402380700735034781635575} a^{6} + \frac{6393993220735935814248839522980922772372311710799700260229}{31006102317414334748740329077370791402380700735034781635575} a^{5} + \frac{1162516898556828207915181812028795342649970151682877313782}{6201220463482866949748065815474158280476140147006956327115} a^{4} - \frac{794187218994249734697697561218379353277831993812125737646}{1823888371612607926396489945727693611904747102060869507975} a^{3} + \frac{2775131092187864836541039596152788510888561679118161753171}{6201220463482866949748065815474158280476140147006956327115} a^{2} - \frac{13478230928094652450659510587674634394899544650805262439932}{31006102317414334748740329077370791402380700735034781635575} a + \frac{2781113058568598551942315973872266302566456228717368385887}{31006102317414334748740329077370791402380700735034781635575}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 14849095216300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SO(3,7)$ (as 21T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $SO(3,7)$
Character table for $SO(3,7)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
107509Data not computed