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Properties

Label 21.21.3907628381...6241.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{32}\cdot 29^{12}$
Root discriminant $132.87$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:(C_7:C_3)$ (as 21T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22707, -8786043, 21272244, 45757012, -204133993, 155541057, 302292949, -737037653, 649246766, -234382540, -30887948, 55589450, -13332074, -2585002, 1538209, -60319, -62440, 6664, 1099, -147, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 147*x^19 + 1099*x^18 + 6664*x^17 - 62440*x^16 - 60319*x^15 + 1538209*x^14 - 2585002*x^13 - 13332074*x^12 + 55589450*x^11 - 30887948*x^10 - 234382540*x^9 + 649246766*x^8 - 737037653*x^7 + 302292949*x^6 + 155541057*x^5 - 204133993*x^4 + 45757012*x^3 + 21272244*x^2 - 8786043*x + 22707)
 
gp: K = bnfinit(x^21 - 7*x^20 - 147*x^19 + 1099*x^18 + 6664*x^17 - 62440*x^16 - 60319*x^15 + 1538209*x^14 - 2585002*x^13 - 13332074*x^12 + 55589450*x^11 - 30887948*x^10 - 234382540*x^9 + 649246766*x^8 - 737037653*x^7 + 302292949*x^6 + 155541057*x^5 - 204133993*x^4 + 45757012*x^3 + 21272244*x^2 - 8786043*x + 22707, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 147 x^{19} + 1099 x^{18} + 6664 x^{17} - 62440 x^{16} - 60319 x^{15} + 1538209 x^{14} - 2585002 x^{13} - 13332074 x^{12} + 55589450 x^{11} - 30887948 x^{10} - 234382540 x^{9} + 649246766 x^{8} - 737037653 x^{7} + 302292949 x^{6} + 155541057 x^{5} - 204133993 x^{4} + 45757012 x^{3} + 21272244 x^{2} - 8786043 x + 22707 \)

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:  \(390762838128733167622487963828338403047536241=7^{32}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.87$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{5}{12} a^{6} - \frac{5}{12} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{14} + \frac{1}{3} a + \frac{1}{4}$, $\frac{1}{24} a^{15} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{5}{24} a^{2} + \frac{3}{8}$, $\frac{1}{24} a^{16} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{5}{24} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{13} - \frac{1}{12} a^{12} - \frac{1}{24} a^{11} - \frac{5}{24} a^{10} + \frac{5}{24} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{7}{24} a^{6} + \frac{1}{12} a^{5} - \frac{1}{24} a^{4} - \frac{5}{12} a^{3} + \frac{1}{24} a^{2} + \frac{1}{12} a - \frac{1}{4}$, $\frac{1}{240} a^{18} - \frac{1}{60} a^{17} - \frac{1}{240} a^{16} - \frac{1}{80} a^{15} - \frac{1}{48} a^{14} + \frac{1}{30} a^{13} + \frac{1}{15} a^{12} - \frac{1}{60} a^{11} + \frac{7}{60} a^{10} + \frac{7}{30} a^{9} + \frac{1}{6} a^{8} - \frac{1}{15} a^{7} + \frac{1}{6} a^{6} - \frac{3}{10} a^{5} + \frac{1}{80} a^{4} + \frac{7}{15} a^{3} + \frac{1}{240} a^{2} + \frac{5}{16} a + \frac{7}{80}$, $\frac{1}{6960} a^{19} + \frac{143}{6960} a^{17} - \frac{17}{6960} a^{16} - \frac{67}{6960} a^{15} + \frac{9}{580} a^{14} + \frac{37}{1740} a^{13} + \frac{65}{696} a^{12} + \frac{71}{3480} a^{11} - \frac{731}{3480} a^{10} - \frac{157}{870} a^{9} - \frac{16}{435} a^{8} + \frac{53}{3480} a^{7} - \frac{181}{580} a^{6} - \frac{451}{1392} a^{5} + \frac{587}{3480} a^{4} - \frac{1621}{6960} a^{3} + \frac{769}{6960} a^{2} - \frac{1229}{6960} a + \frac{1}{40}$, $\frac{1}{11770313400137700869416829753176302561888657939380880} a^{20} + \frac{194085238805498427650323836766171743669263788207}{5885156700068850434708414876588151280944328969690440} a^{19} - \frac{3182442369288177726473992341020516801539980446117}{3923437800045900289805609917725434187296219313126960} a^{18} + \frac{9256954635785335437665622574161501933719768711251}{11770313400137700869416829753176302561888657939380880} a^{17} + \frac{134475830580816190350778745629685514789867477839089}{11770313400137700869416829753176302561888657939380880} a^{16} + \frac{22383929860997249064542954897143083408026872659983}{2942578350034425217354207438294075640472164484845220} a^{15} + \frac{20383680922603798276172297647273977680410532828207}{1177031340013770086941682975317630256188865793938088} a^{14} - \frac{1870371319836850236405507667017995904369530157455}{294257835003442521735420743829407564047216448484522} a^{13} + \frac{301081345126024320739326077532202687305966612503089}{5885156700068850434708414876588151280944328969690440} a^{12} + \frac{38866105047565992293698123588464977447086176178627}{735644587508606304338551859573518910118041121211305} a^{11} + \frac{1412714254556049273339382940016031669048212117480247}{5885156700068850434708414876588151280944328969690440} a^{10} + \frac{260653787368204975381745495307117304748356159994813}{5885156700068850434708414876588151280944328969690440} a^{9} - \frac{1118691975461023122028713385918085123166350944356869}{5885156700068850434708414876588151280944328969690440} a^{8} + \frac{205906477741240955839956525308987888083773415478137}{1471289175017212608677103719147037820236082242422610} a^{7} + \frac{2189330121393830381870281895645371205253944466457567}{11770313400137700869416829753176302561888657939380880} a^{6} - \frac{556176367522018622238872007918090896953738258660247}{2942578350034425217354207438294075640472164484845220} a^{5} - \frac{1206450924753558697694299659127479153070125880853899}{3923437800045900289805609917725434187296219313126960} a^{4} - \frac{5000018912712510415169060607600481881005332539087253}{11770313400137700869416829753176302561888657939380880} a^{3} - \frac{99123803829562162081780095935275408524212052758207}{11770313400137700869416829753176302561888657939380880} a^{2} + \frac{14871550468140052941619431377521516917406636792361}{653906300007650048300934986287572364549369885521160} a + \frac{8926127067320200463030115260627202339902427475141}{22548493103712070631066723665088702225840340880040}$

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:  \( 13788563081500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:(C_7:C_3)$ (as 21T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147
The 19 conjugacy class representatives for $C_7:(C_7:C_3)$
Character table for $C_7:(C_7:C_3)$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 21 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$