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Properties

Label 21.3.20301208447...9375.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{28}\cdot 5^{18}\cdot 7^{17}$
Root discriminant $83.06$
Ramified primes $3, 5, 7$
Class number $42$ (GRH)
Class group $[42]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18017937, 50031891, -121483152, 35144488, 47557968, -37483542, -10608182, 7030128, -510537, -1479827, 451815, 241392, -9420, -28155, -5685, 2889, 648, -6, -96, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^19 - 96*x^18 - 6*x^17 + 648*x^16 + 2889*x^15 - 5685*x^14 - 28155*x^13 - 9420*x^12 + 241392*x^11 + 451815*x^10 - 1479827*x^9 - 510537*x^8 + 7030128*x^7 - 10608182*x^6 - 37483542*x^5 + 47557968*x^4 + 35144488*x^3 - 121483152*x^2 + 50031891*x - 18017937)
 
gp: K = bnfinit(x^21 - 6*x^19 - 96*x^18 - 6*x^17 + 648*x^16 + 2889*x^15 - 5685*x^14 - 28155*x^13 - 9420*x^12 + 241392*x^11 + 451815*x^10 - 1479827*x^9 - 510537*x^8 + 7030128*x^7 - 10608182*x^6 - 37483542*x^5 + 47557968*x^4 + 35144488*x^3 - 121483152*x^2 + 50031891*x - 18017937, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{19} - 96 x^{18} - 6 x^{17} + 648 x^{16} + 2889 x^{15} - 5685 x^{14} - 28155 x^{13} - 9420 x^{12} + 241392 x^{11} + 451815 x^{10} - 1479827 x^{9} - 510537 x^{8} + 7030128 x^{7} - 10608182 x^{6} - 37483542 x^{5} + 47557968 x^{4} + 35144488 x^{3} - 121483152 x^{2} + 50031891 x - 18017937 \)

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:  \(-20301208447174974342642532070159912109375=-\,3^{28}\cdot 5^{18}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.06$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{15} a^{19} - \frac{1}{5} a^{9} - \frac{1}{3} a^{7} + \frac{7}{15} a^{4} - \frac{1}{3} a$, $\frac{1}{385619483900987517152857856963964343309923646213034337293522706823245972351086355} a^{20} - \frac{431655275335845306845283711960197323910637352503136655979639858523222388086619}{18362832566713691292993231283998302062377316486334968442548700324916474873861255} a^{19} + \frac{1039565246983396606301241609608870611400742253749881800291218592730533216982276}{25707965593399167810190523797597622887328243080868955819568180454883064823405757} a^{18} - \frac{169786651701584140111194602596091482361266536449654700917004654691812165418613}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{17} + \frac{905029512190520372521460565872296858663418071458405023020422066932760998528659}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{16} - \frac{2000553526530169590210186939684339503263821809819629588638373888255292700597862}{42846609322331946350317539662662704812213738468114926365946967424805108039009595} a^{15} - \frac{729757626212411235639696224813464128949802577369279199400275023359144656145364}{8569321864466389270063507932532540962442747693622985273189393484961021607801919} a^{14} + \frac{1167235119269576536342462652038574699659725321397260761364993604797360378887144}{11685438906090530822813874453453464948785565036758616281621900206765029465184435} a^{13} - \frac{5999285320911040852544091681034101631868244467178278623092846869595642754408636}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{12} + \frac{10071572661314368346627246204131050326621794750755667535067284644131388278969808}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{11} - \frac{37671718979916977433614500253202402778417453203045498625475504724393849892330211}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{10} - \frac{3224928377954553791063325991787075061335291634966633571292909619050020284241072}{18362832566713691292993231283998302062377316486334968442548700324916474873861255} a^{9} - \frac{26245509692542395719664704404932956105896226220816675370339207357890901793491015}{77123896780197503430571571392792868661984729242606867458704541364649194470217271} a^{8} + \frac{167922382498911056978958891789052065324980198485739326438598712167486094238442}{42846609322331946350317539662662704812213738468114926365946967424805108039009595} a^{7} + \frac{632538292854872961672148664361764766493379678906292827391895270880478488963531}{3672566513342738258598646256799660412475463297266993688509740064983294974772251} a^{6} - \frac{137378979624851254147208679265757012624246380936331209933770383605496648687646143}{385619483900987517152857856963964343309923646213034337293522706823245972351086355} a^{5} - \frac{57807151431343469059449392859457666384984862889151043860451157379333301976478523}{128539827966995839050952618987988114436641215404344779097840902274415324117028785} a^{4} - \frac{2810143894961193498342279580032915930310919076191268153924000893626226773978829}{11685438906090530822813874453453464948785565036758616281621900206765029465184435} a^{3} + \frac{112443406070225653467664601094490594442645132514110205879310128571408940874167359}{385619483900987517152857856963964343309923646213034337293522706823245972351086355} a^{2} + \frac{1208048932582991740244612461063714024685438989292694291432636739052579559077299}{18362832566713691292993231283998302062377316486334968442548700324916474873861255} a - \frac{757522804035797095739763603469810658195234449489249351386687084874009807299979}{6120944188904563764331077094666100687459105495444989480849566774972158291287085}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{42}$, which has order $42$ (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:  \( 37226343333.12808 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.3969.2, 7.1.1722980109375.2

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 $21$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
3Data not computed
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
7Data not computed