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Properties

Label 21.21.5390245643...7056.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 728809^{5}$
Root discriminant $45.07$
Ramified primes $2, 728809$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T38

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 23, 194, -352, -2351, 2191, 10604, -6770, -22602, 11568, 25300, -11326, -15522, 6380, 5248, -2026, -953, 349, 86, -30, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 30*x^19 + 86*x^18 + 349*x^17 - 953*x^16 - 2026*x^15 + 5248*x^14 + 6380*x^13 - 15522*x^12 - 11326*x^11 + 25300*x^10 + 11568*x^9 - 22602*x^8 - 6770*x^7 + 10604*x^6 + 2191*x^5 - 2351*x^4 - 352*x^3 + 194*x^2 + 23*x - 1)
 
gp: K = bnfinit(x^21 - 3*x^20 - 30*x^19 + 86*x^18 + 349*x^17 - 953*x^16 - 2026*x^15 + 5248*x^14 + 6380*x^13 - 15522*x^12 - 11326*x^11 + 25300*x^10 + 11568*x^9 - 22602*x^8 - 6770*x^7 + 10604*x^6 + 2191*x^5 - 2351*x^4 - 352*x^3 + 194*x^2 + 23*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 30 x^{19} + 86 x^{18} + 349 x^{17} - 953 x^{16} - 2026 x^{15} + 5248 x^{14} + 6380 x^{13} - 15522 x^{12} - 11326 x^{11} + 25300 x^{10} + 11568 x^{9} - 22602 x^{8} - 6770 x^{7} + 10604 x^{6} + 2191 x^{5} - 2351 x^{4} - 352 x^{3} + 194 x^{2} + 23 x - 1 \)

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:  \(53902456438168738728940914094637056=2^{18}\cdot 728809^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.07$
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:  $2, 728809$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{16} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{3}{8} a^{3} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{1301337888175022344} a^{20} - \frac{4695771512853955}{650668944087511172} a^{19} + \frac{19024413221282141}{325334472043755586} a^{18} + \frac{2801051185226524}{162667236021877793} a^{17} + \frac{87755288420667091}{1301337888175022344} a^{16} - \frac{70382622588435193}{650668944087511172} a^{15} + \frac{14205809100110081}{162667236021877793} a^{14} - \frac{38840832489615351}{325334472043755586} a^{13} + \frac{23765069301457497}{325334472043755586} a^{12} - \frac{16317489162781651}{650668944087511172} a^{11} + \frac{16892781939281067}{325334472043755586} a^{10} - \frac{18051876364282605}{325334472043755586} a^{9} + \frac{3376680406535993}{162667236021877793} a^{8} + \frac{44413907333473525}{650668944087511172} a^{7} + \frac{18993296439154574}{162667236021877793} a^{6} + \frac{107581827888441595}{325334472043755586} a^{5} + \frac{305599737318537995}{1301337888175022344} a^{4} - \frac{131249323672258651}{325334472043755586} a^{3} - \frac{65774852812051674}{162667236021877793} a^{2} + \frac{60784069746797613}{325334472043755586} a - \frac{635877871975567299}{1301337888175022344}$

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:  \( 47133176363.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T38:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for t21n38
Character table for t21n38

Intermediate fields

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

Sibling fields

Degree 7 sibling: 7.7.46643776.1
Degree 14 sibling: Deg 14
Degree 30 sibling: data not computed
Degree 35 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
728809Data not computed