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Properties

Label 21.21.7702397160...2736.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{6}\cdot 7^{14}\cdot 13^{2}\cdot 43^{2}\cdot 127^{2}\cdot 9955541^{2}\cdot 59601376927^{2}$
Root discriminant $636.98$
Ramified primes $2, 7, 13, 43, 127, 9955541, 59601376927$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T153

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2113, 35908, -220083, 141870, 1176607, -2545651, 800599, 2510694, -2348678, -384952, 1207545, -203817, -279847, 89028, 34478, -14856, -2311, 1275, 78, -56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 56*x^19 + 78*x^18 + 1275*x^17 - 2311*x^16 - 14856*x^15 + 34478*x^14 + 89028*x^13 - 279847*x^12 - 203817*x^11 + 1207545*x^10 - 384952*x^9 - 2348678*x^8 + 2510694*x^7 + 800599*x^6 - 2545651*x^5 + 1176607*x^4 + 141870*x^3 - 220083*x^2 + 35908*x + 2113)
 
gp: K = bnfinit(x^21 - x^20 - 56*x^19 + 78*x^18 + 1275*x^17 - 2311*x^16 - 14856*x^15 + 34478*x^14 + 89028*x^13 - 279847*x^12 - 203817*x^11 + 1207545*x^10 - 384952*x^9 - 2348678*x^8 + 2510694*x^7 + 800599*x^6 - 2545651*x^5 + 1176607*x^4 + 141870*x^3 - 220083*x^2 + 35908*x + 2113, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 56 x^{19} + 78 x^{18} + 1275 x^{17} - 2311 x^{16} - 14856 x^{15} + 34478 x^{14} + 89028 x^{13} - 279847 x^{12} - 203817 x^{11} + 1207545 x^{10} - 384952 x^{9} - 2348678 x^{8} + 2510694 x^{7} + 800599 x^{6} - 2545651 x^{5} + 1176607 x^{4} + 141870 x^{3} - 220083 x^{2} + 35908 x + 2113 \)

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:  \(77023971602373143047450273918178286388112224004493606492736=2^{6}\cdot 7^{14}\cdot 13^{2}\cdot 43^{2}\cdot 127^{2}\cdot 9955541^{2}\cdot 59601376927^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $636.98$
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:  $2, 7, 13, 43, 127, 9955541, 59601376927$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{4}{13} a^{17} - \frac{1}{13} a^{16} - \frac{2}{13} a^{14} - \frac{2}{13} a^{13} + \frac{1}{13} a^{12} - \frac{1}{13} a^{11} + \frac{5}{13} a^{10} + \frac{2}{13} a^{9} + \frac{5}{13} a^{8} + \frac{6}{13} a^{7} + \frac{4}{13} a^{6} + \frac{3}{13} a^{5} - \frac{1}{13} a^{4} - \frac{4}{13} a^{3} + \frac{4}{13} a^{2} + \frac{5}{13} a + \frac{4}{13}$, $\frac{1}{2978321476} a^{19} + \frac{26278653}{2978321476} a^{18} - \frac{1410594847}{2978321476} a^{17} - \frac{229585649}{2978321476} a^{16} - \frac{304045216}{744580369} a^{15} + \frac{496375417}{1489160738} a^{14} - \frac{19498112}{57275413} a^{13} + \frac{91342272}{744580369} a^{12} + \frac{176568255}{744580369} a^{11} + \frac{963657433}{2978321476} a^{10} + \frac{400369521}{2978321476} a^{9} - \frac{675585585}{1489160738} a^{8} - \frac{468051161}{2978321476} a^{7} + \frac{505573675}{1489160738} a^{6} + \frac{1005717291}{2978321476} a^{5} - \frac{1167451361}{2978321476} a^{4} - \frac{105932222}{744580369} a^{3} - \frac{330890022}{744580369} a^{2} - \frac{4321955}{114550826} a + \frac{1264311141}{2978321476}$, $\frac{1}{8870398814402818576} a^{20} + \frac{6569663}{4435199407201409288} a^{19} + \frac{6640073110521}{341169185169339176} a^{18} - \frac{275797338190147}{1009376287483252} a^{17} - \frac{2905347684770636793}{8870398814402818576} a^{16} - \frac{1008192647052446455}{4435199407201409288} a^{15} + \frac{887222159513788003}{4435199407201409288} a^{14} + \frac{444403150350121395}{1108799851800352322} a^{13} - \frac{281742100996894281}{2217599703600704644} a^{12} - \frac{1266611451442398467}{8870398814402818576} a^{11} - \frac{773417043490294363}{4435199407201409288} a^{10} - \frac{4377691053443048401}{8870398814402818576} a^{9} + \frac{577457285778930825}{8870398814402818576} a^{8} - \frac{1964303401885098415}{8870398814402818576} a^{7} - \frac{1870235779471976251}{8870398814402818576} a^{6} - \frac{124619249102342703}{4435199407201409288} a^{5} + \frac{927793844494691723}{8870398814402818576} a^{4} - \frac{594807722313782443}{2217599703600704644} a^{3} + \frac{744418191231023925}{4435199407201409288} a^{2} + \frac{664519099052450995}{8870398814402818576} a - \frac{3831993881814367775}{8870398814402818576}$

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:  \( 134278838041000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T153:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 48009024000
The 267 conjugacy class representatives for t21n153 are not computed
Character table for t21n153 is not computed

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 45 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ $21$ $15{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ $15{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $21$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.7.0.1$x^{7} - 2 x + 9$$1$$7$$0$$C_7$$[\ ]^{7}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.3.2.3$x^{3} - 10287$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
127.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
9955541Data not computed
59601376927Data not computed