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Properties

Label 21.1.96191904229...3632.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 7^{21}\cdot 29^{7}$
Root discriminant $172.70$
Ramified primes $2, 3, 7, 29$
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![348528, -1193808, 1010352, -677040, 2003568, -1988448, 1927968, -1702904, 564704, 47152, -16100, -49252, 36862, -6174, -5271, 2793, -35, -343, 91, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 7*x^19 + 91*x^18 - 343*x^17 - 35*x^16 + 2793*x^15 - 5271*x^14 - 6174*x^13 + 36862*x^12 - 49252*x^11 - 16100*x^10 + 47152*x^9 + 564704*x^8 - 1702904*x^7 + 1927968*x^6 - 1988448*x^5 + 2003568*x^4 - 677040*x^3 + 1010352*x^2 - 1193808*x + 348528)
 
gp: K = bnfinit(x^21 - 7*x^20 + 7*x^19 + 91*x^18 - 343*x^17 - 35*x^16 + 2793*x^15 - 5271*x^14 - 6174*x^13 + 36862*x^12 - 49252*x^11 - 16100*x^10 + 47152*x^9 + 564704*x^8 - 1702904*x^7 + 1927968*x^6 - 1988448*x^5 + 2003568*x^4 - 677040*x^3 + 1010352*x^2 - 1193808*x + 348528, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 7 x^{19} + 91 x^{18} - 343 x^{17} - 35 x^{16} + 2793 x^{15} - 5271 x^{14} - 6174 x^{13} + 36862 x^{12} - 49252 x^{11} - 16100 x^{10} + 47152 x^{9} + 564704 x^{8} - 1702904 x^{7} + 1927968 x^{6} - 1988448 x^{5} + 2003568 x^{4} - 677040 x^{3} + 1010352 x^{2} - 1193808 x + 348528 \)

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:  \(96191904229042724857909956685420452170151493632=2^{33}\cdot 3^{19}\cdot 7^{21}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.70$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{4} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{15} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{16} - \frac{1}{4} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{16} - \frac{1}{24} a^{15} - \frac{1}{24} a^{14} - \frac{1}{8} a^{13} + \frac{1}{24} a^{12} + \frac{1}{24} a^{11} - \frac{1}{8} a^{10} + \frac{1}{12} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{24} a^{18} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{72} a^{19} - \frac{1}{36} a^{16} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{24} a^{11} - \frac{1}{9} a^{10} + \frac{1}{12} a^{8} + \frac{11}{36} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{26245960113453095159635953535287415089103475148326702472} a^{20} + \frac{1061923277999976666211695614655403322428113322065253}{224324445414129018458426953293054829821397223489971816} a^{19} - \frac{13029027561180194118637665502403321606628730930921369}{4374326685575515859939325589214569181517245858054450412} a^{18} - \frac{20086260784250928413833022346590137793707518018983195}{26245960113453095159635953535287415089103475148326702472} a^{17} - \frac{18788627006041137061004442458451582127964180211270971}{672973336242387055375280859879164489464191670469915448} a^{16} + \frac{213588023001589131795867337882785239343065025698009737}{8748653371151031719878651178429138363034491716108900824} a^{15} - \frac{7874524000400398117604514718695967115319392433594795}{224324445414129018458426953293054829821397223489971816} a^{14} + \frac{1962124295884225453186415063181756213196796207530490717}{8748653371151031719878651178429138363034491716108900824} a^{13} - \frac{266055783849141033750582321933250054315343335942567066}{1093581671393878964984831397303642295379311464513612603} a^{12} - \frac{666220250751449164375317964446302638766215529835149041}{3280745014181636894954494191910926886137934393540837809} a^{11} - \frac{1140616475918541140473862564834586389159451528467125141}{8748653371151031719878651178429138363034491716108900824} a^{10} + \frac{17680035020363672101018853677910104881761925626797142}{1093581671393878964984831397303642295379311464513612603} a^{9} - \frac{42380781637342441068091474942833251067003480086734891}{252365001090895145765730322454686683549071876426218293} a^{8} + \frac{13294471071671628914291441246824651960179103353090403}{37709712806685481551201082665642837771700395328055607} a^{7} + \frac{19522894643699897144152801673710187832599133719758079}{729054447595919309989887598202428196919540976342408402} a^{6} + \frac{300606550380916668514809326026550104230205774149192865}{729054447595919309989887598202428196919540976342408402} a^{5} + \frac{433822569450967629553503332743452333308150334635934092}{1093581671393878964984831397303642295379311464513612603} a^{4} - \frac{14596220254746059225562395907352552914186483079685279}{56081111353532254614606738323263707455349305872492954} a^{3} + \frac{4545354044873457419988133152570810362735179756427913}{28040555676766127307303369161631853727674652936246477} a^{2} + \frac{465534840182602015523695827317010009029502821096855390}{1093581671393878964984831397303642295379311464513612603} a - \frac{42046828225196397338403706184395262644404702311413061}{364527223797959654994943799101214098459770488171204201}$

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:  \( 9623787853487828.0 \) (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

3.1.696.1, 7.1.38423222208.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
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$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$