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Properties

Label 21.1.28075350130...5625.1
Degree $21$
Signature $[1, 10]$
Discriminant $5^{14}\cdot 7^{28}$
Root discriminant $39.15$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{21}$ (as 21T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-405, 945, -1575, 4389, -12320, 26600, -39067, 39987, -32417, 21700, -16709, 12166, -7224, 3360, -1355, 693, -490, 266, -112, 35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405)
 
gp: K = bnfinit(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 35 x^{19} - 112 x^{18} + 266 x^{17} - 490 x^{16} + 693 x^{15} - 1355 x^{14} + 3360 x^{13} - 7224 x^{12} + 12166 x^{11} - 16709 x^{10} + 21700 x^{9} - 32417 x^{8} + 39987 x^{7} - 39067 x^{6} + 26600 x^{5} - 12320 x^{4} + 4389 x^{3} - 1575 x^{2} + 945 x - 405 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2807535013090453863383795166015625=5^{14}\cdot 7^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.15$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{63} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{8} + \frac{1}{7} a^{7} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{3}{7}$, $\frac{1}{315} a^{15} + \frac{1}{15} a^{13} + \frac{1}{45} a^{12} - \frac{1}{45} a^{11} - \frac{1}{45} a^{10} + \frac{52}{105} a^{8} - \frac{19}{45} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{16}{45} a^{4} + \frac{19}{45} a^{3} + \frac{2}{9} a^{2} - \frac{2}{7} a$, $\frac{1}{315} a^{16} + \frac{1}{315} a^{14} - \frac{4}{45} a^{13} - \frac{2}{15} a^{12} + \frac{4}{45} a^{11} - \frac{1}{9} a^{10} + \frac{17}{105} a^{9} - \frac{4}{45} a^{8} + \frac{2}{21} a^{7} + \frac{1}{9} a^{6} + \frac{4}{45} a^{5} + \frac{1}{5} a^{4} + \frac{4}{9} a^{3} + \frac{10}{63} a^{2} - \frac{1}{3} a - \frac{2}{7}$, $\frac{1}{315} a^{17} + \frac{2}{315} a^{14} + \frac{2}{15} a^{13} + \frac{1}{15} a^{12} - \frac{4}{45} a^{11} - \frac{47}{315} a^{10} - \frac{4}{45} a^{9} - \frac{2}{5} a^{8} + \frac{41}{105} a^{7} - \frac{11}{45} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{83}{315} a^{3} - \frac{2}{9} a^{2} + \frac{3}{7}$, $\frac{1}{945} a^{18} + \frac{1}{945} a^{17} - \frac{1}{945} a^{16} + \frac{1}{945} a^{15} + \frac{1}{315} a^{14} - \frac{1}{9} a^{13} + \frac{1}{15} a^{12} - \frac{26}{945} a^{11} - \frac{103}{945} a^{10} + \frac{22}{189} a^{9} - \frac{236}{945} a^{8} + \frac{419}{945} a^{7} + \frac{1}{45} a^{6} + \frac{7}{45} a^{5} + \frac{32}{105} a^{4} + \frac{344}{945} a^{3} + \frac{2}{21} a^{2} + \frac{5}{21} a - \frac{2}{7}$, $\frac{1}{208845} a^{19} - \frac{2}{4641} a^{18} - \frac{311}{208845} a^{17} + \frac{62}{208845} a^{16} + \frac{40}{41769} a^{15} + \frac{146}{23205} a^{14} - \frac{304}{3315} a^{13} + \frac{34498}{208845} a^{12} - \frac{4541}{208845} a^{11} - \frac{31}{273} a^{10} + \frac{17666}{208845} a^{9} + \frac{69742}{208845} a^{8} + \frac{101356}{208845} a^{7} - \frac{44}{255} a^{6} + \frac{5248}{69615} a^{5} + \frac{87977}{208845} a^{4} + \frac{44377}{208845} a^{3} - \frac{4187}{13923} a^{2} - \frac{25}{91} a + \frac{90}{1547}$, $\frac{1}{23838698252740166694854865} a^{20} + \frac{2588398102915854092}{1254668329091587720781835} a^{19} + \frac{180761284815865357654}{507206345802982270103295} a^{18} + \frac{33547081259788491149}{1254668329091587720781835} a^{17} + \frac{381174600435484865920}{250933665818317544156367} a^{16} - \frac{4524603732312943752238}{3405528321820023813550695} a^{15} + \frac{50699500859660353075702}{7946232750913388898284955} a^{14} - \frac{1607669055243318205613}{116286332940195935096853} a^{13} + \frac{141375327629758138816457}{882914750101487655364995} a^{12} - \frac{57881991994445202423386}{611248673147183761406535} a^{11} + \frac{1937801545059915629476666}{23838698252740166694854865} a^{10} - \frac{440274597039071444243411}{23838698252740166694854865} a^{9} + \frac{1089062695031138790536341}{3405528321820023813550695} a^{8} - \frac{373384733848599719469283}{1402276367808245099697345} a^{7} + \frac{3546523592497731582090233}{7946232750913388898284955} a^{6} + \frac{479374250360843454625921}{4767739650548033338970973} a^{5} + \frac{3734133859793052408131423}{23838698252740166694854865} a^{4} + \frac{83029382772190434842044}{1254668329091587720781835} a^{3} - \frac{58290527673636075494}{2040111104213963773629} a^{2} + \frac{28032487093250014087364}{75678407151556084745571} a + \frac{71141271643077812158259}{176582950020297531072999}$

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:  $10$
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:  \( 859506502.583 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{21}$ (as 21T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 12 conjugacy class representatives for $D_{21}$
Character table for $D_{21}$

Intermediate fields

3.1.175.1, 7.1.40353607.1

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

Sibling fields

Galois closure: 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 $21$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R R $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
7Data not computed