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Properties

Label 21.3.44553216873...7527.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{48}\cdot 7^{21}$
Root discriminant $86.23$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T42

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-443448, 72576, -283374, 562625, 563472, 31752, 7644, -396243, -14994, 80388, -33516, 11151, 10192, 3528, 1224, -1008, 252, 63, -42, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 42*x^18 + 63*x^17 + 252*x^16 - 1008*x^15 + 1224*x^14 + 3528*x^13 + 10192*x^12 + 11151*x^11 - 33516*x^10 + 80388*x^9 - 14994*x^8 - 396243*x^7 + 7644*x^6 + 31752*x^5 + 563472*x^4 + 562625*x^3 - 283374*x^2 + 72576*x - 443448)
 
gp: K = bnfinit(x^21 - 42*x^18 + 63*x^17 + 252*x^16 - 1008*x^15 + 1224*x^14 + 3528*x^13 + 10192*x^12 + 11151*x^11 - 33516*x^10 + 80388*x^9 - 14994*x^8 - 396243*x^7 + 7644*x^6 + 31752*x^5 + 563472*x^4 + 562625*x^3 - 283374*x^2 + 72576*x - 443448, 1)
 

Normalized defining polynomial

\( x^{21} - 42 x^{18} + 63 x^{17} + 252 x^{16} - 1008 x^{15} + 1224 x^{14} + 3528 x^{13} + 10192 x^{12} + 11151 x^{11} - 33516 x^{10} + 80388 x^{9} - 14994 x^{8} - 396243 x^{7} + 7644 x^{6} + 31752 x^{5} + 563472 x^{4} + 562625 x^{3} - 283374 x^{2} + 72576 x - 443448 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44553216873221843443058843338471726567527=-\,3^{48}\cdot 7^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.23$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{19} + \frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{46064693399184196907334709806843628820117594765413400358151572} a^{20} - \frac{178313205830635105307737964309475323745610547980408434878395}{2559149633288010939296372767046868267784310820300744464341754} a^{19} + \frac{389261167835450818542520193976052020605141156343824850814691}{7677448899864032817889118301140604803352932460902233393025262} a^{18} - \frac{214104838173815056532537236383093312148169353212653135762828}{3838724449932016408944559150570302401676466230451116696512631} a^{17} - \frac{656998781888910356931731466510443735591138871063066954615749}{15354897799728065635778236602281209606705864921804466786050524} a^{16} + \frac{75704784354163121592292347696475333288843660711517073945127}{2559149633288010939296372767046868267784310820300744464341754} a^{15} + \frac{501831617203508407172799396185807968776541706050664026192115}{7677448899864032817889118301140604803352932460902233393025262} a^{14} + \frac{263995353036476337756553747975580946298542313704265842210691}{7677448899864032817889118301140604803352932460902233393025262} a^{13} - \frac{109237996657529938676966566612065370390544215904634672907983}{2559149633288010939296372767046868267784310820300744464341754} a^{12} + \frac{129872819247515387956392713251866003802958999001063774143992}{11516173349796049226833677451710907205029398691353350089537893} a^{11} + \frac{495316443507407410859525552086872417744217308364322474602301}{5118299266576021878592745534093736535568621640601488928683508} a^{10} + \frac{3284051895892647559789353965476458493038589381529229738934529}{7677448899864032817889118301140604803352932460902233393025262} a^{9} + \frac{1779441449733446903021182177784640267539815686695321314252303}{7677448899864032817889118301140604803352932460902233393025262} a^{8} - \frac{3809896898708817797051199480496700167381357978015231368550301}{7677448899864032817889118301140604803352932460902233393025262} a^{7} + \frac{6668957153092082583492623791624497044661892321244267630229335}{15354897799728065635778236602281209606705864921804466786050524} a^{6} + \frac{1221470529346440255298465100797495614079047980210385235693739}{2559149633288010939296372767046868267784310820300744464341754} a^{5} - \frac{222811924687406484569725686472917956941106295198657052100057}{2559149633288010939296372767046868267784310820300744464341754} a^{4} + \frac{409944231518784078870308784718707495314311763303134967166649}{7677448899864032817889118301140604803352932460902233393025262} a^{3} + \frac{18592331727863972212822668539458162084444940494704797600453321}{46064693399184196907334709806843628820117594765413400358151572} a^{2} + \frac{364390891566517235281228298685658979527864344960325153504498}{1279574816644005469648186383523434133892155410150372232170877} a - \frac{557078635189768187809013901083929869105290235192604468186560}{1279574816644005469648186383523434133892155410150372232170877}$

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:  $11$
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:  \( 1357254430200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T42:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6174
The 37 conjugacy class representatives for t21n42
Character table for t21n42 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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

Sibling fields

Degree 21 siblings: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R $18{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $18{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $18{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
3Data not computed
7Data not computed