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Properties

Label 21.7.41910614826...0016.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,2^{30}\cdot 3^{27}\cdot 13^{15}$
Root discriminant $69.05$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1646714, 4802307, -6344232, -6027167, 2869290, -396504, 237822, 143181, -240930, 134624, -31386, 37740, -27152, 5994, -72, 993, -756, 117, 14, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 6*x^19 + 14*x^18 + 117*x^17 - 756*x^16 + 993*x^15 - 72*x^14 + 5994*x^13 - 27152*x^12 + 37740*x^11 - 31386*x^10 + 134624*x^9 - 240930*x^8 + 143181*x^7 + 237822*x^6 - 396504*x^5 + 2869290*x^4 - 6027167*x^3 - 6344232*x^2 + 4802307*x + 1646714)
 
gp: K = bnfinit(x^21 - 6*x^20 + 6*x^19 + 14*x^18 + 117*x^17 - 756*x^16 + 993*x^15 - 72*x^14 + 5994*x^13 - 27152*x^12 + 37740*x^11 - 31386*x^10 + 134624*x^9 - 240930*x^8 + 143181*x^7 + 237822*x^6 - 396504*x^5 + 2869290*x^4 - 6027167*x^3 - 6344232*x^2 + 4802307*x + 1646714, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} + 6 x^{19} + 14 x^{18} + 117 x^{17} - 756 x^{16} + 993 x^{15} - 72 x^{14} + 5994 x^{13} - 27152 x^{12} + 37740 x^{11} - 31386 x^{10} + 134624 x^{9} - 240930 x^{8} + 143181 x^{7} + 237822 x^{6} - 396504 x^{5} + 2869290 x^{4} - 6027167 x^{3} - 6344232 x^{2} + 4802307 x + 1646714 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-419106148260412830400407432145473110016=-\,2^{30}\cdot 3^{27}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.05$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{6} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{18} a^{4} - \frac{4}{9} a^{3} + \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{18} a^{5} - \frac{1}{6} a^{4} + \frac{7}{18} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9}$, $\frac{1}{72} a^{15} - \frac{1}{36} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{18} a^{6} - \frac{1}{18} a^{5} - \frac{1}{3} a^{4} - \frac{5}{12} a^{3} + \frac{1}{18} a^{2} + \frac{1}{24} a + \frac{17}{36}$, $\frac{1}{72} a^{16} + \frac{1}{18} a^{12} - \frac{1}{6} a^{11} - \frac{1}{18} a^{10} + \frac{1}{36} a^{9} + \frac{1}{12} a^{8} - \frac{1}{18} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{18} a^{3} + \frac{3}{8} a^{2} - \frac{1}{9} a - \frac{5}{18}$, $\frac{1}{72} a^{17} + \frac{1}{18} a^{12} + \frac{1}{9} a^{11} + \frac{1}{12} a^{10} + \frac{1}{36} a^{9} - \frac{1}{18} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{12} a^{5} + \frac{1}{3} a^{4} - \frac{1}{72} a^{3} - \frac{5}{18} a^{2} - \frac{1}{2} a - \frac{1}{9}$, $\frac{1}{144} a^{18} - \frac{1}{144} a^{16} - \frac{1}{144} a^{15} - \frac{1}{72} a^{14} - \frac{11}{72} a^{11} + \frac{11}{72} a^{10} + \frac{7}{72} a^{9} + \frac{1}{12} a^{8} + \frac{11}{72} a^{7} - \frac{7}{72} a^{6} + \frac{17}{36} a^{5} - \frac{5}{48} a^{4} - \frac{7}{24} a^{3} - \frac{47}{144} a^{2} - \frac{19}{144} a + \frac{7}{24}$, $\frac{1}{432} a^{19} - \frac{1}{432} a^{18} - \frac{1}{144} a^{17} - \frac{1}{144} a^{15} + \frac{1}{72} a^{14} + \frac{1}{24} a^{12} + \frac{5}{36} a^{11} - \frac{5}{108} a^{10} - \frac{11}{216} a^{9} + \frac{5}{72} a^{8} + \frac{23}{72} a^{6} + \frac{19}{144} a^{5} + \frac{13}{48} a^{4} + \frac{17}{48} a^{3} + \frac{1}{36} a^{2} - \frac{17}{432} a - \frac{47}{216}$, $\frac{1}{81725192352984577191120201795663291337139268914906960592} a^{20} + \frac{70249854173293295521332128804756217651593102073456491}{81725192352984577191120201795663291337139268914906960592} a^{19} + \frac{15765577249417351261346695451644881533933654195826961}{9080576928109397465680022421740365704126585434989662288} a^{18} + \frac{43126953623828991650838060418072513015786770065795821}{6810432696082048099260016816305274278094939076242246716} a^{17} + \frac{5253578240867777405119439731409662656685468514077905}{3026858976036465821893340807246788568042195144996554096} a^{16} - \frac{22049560387218827665086415047801984839545962451594165}{4540288464054698732840011210870182852063292717494831144} a^{15} + \frac{49243809696403726215904634187926349821723706480224605}{3405216348041024049630008408152637139047469538121123358} a^{14} - \frac{14467775798579201605127164539668667290265824314070017}{13620865392164096198520033632610548556189878152484493432} a^{13} - \frac{79523356743329777418948526421363997258641594615307193}{6810432696082048099260016816305274278094939076242246716} a^{12} - \frac{1087654193698395845986064374641547082508199786861826555}{20431298088246144297780050448915822834284817228726740148} a^{11} - \frac{2090073976860077404958523428673030212099582227886784359}{40862596176492288595560100897831645668569634457453480296} a^{10} + \frac{1353965369490483959261485077837374428026797891115850991}{13620865392164096198520033632610548556189878152484493432} a^{9} + \frac{475246523047817239841914638933610677725270286945852239}{6810432696082048099260016816305274278094939076242246716} a^{8} - \frac{159125312420272298767300097212646316270571692724755887}{4540288464054698732840011210870182852063292717494831144} a^{7} + \frac{1018999284980730726403565168762747713407631072969682547}{3026858976036465821893340807246788568042195144996554096} a^{6} - \frac{1918694623333619974535754035411829562516983782917018181}{27241730784328192397040067265221097112379756304968986864} a^{5} + \frac{10670616457557838677396248620251055776794948486474258219}{27241730784328192397040067265221097112379756304968986864} a^{4} - \frac{256973837760001386006992621452354648452727434470939473}{3405216348041024049630008408152637139047469538121123358} a^{3} + \frac{21849301720442310267587024473441494793594244148408317569}{81725192352984577191120201795663291337139268914906960592} a^{2} + \frac{15589039777665087704844531422802282310240111321549949329}{40862596176492288595560100897831645668569634457453480296} a + \frac{91121838642656031382145848446251201551030876524207517}{3405216348041024049630008408152637139047469538121123358}$

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:  $13$
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:  \( 1564637139010 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.216.1, 7.7.138584369664.1

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

Sibling fields

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 R $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$