↑

Properties

Label 21.3.99050649772...9248.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 127^{7}$
Root discriminant $268.15$
Ramified primes $2, 3, 7, 127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7854656, 9202368, -9240448, -4262720, 2944032, 2329152, -1194704, 650208, -667016, -63896, 167748, 153188, -120750, -22890, 25169, 1701, -3255, 77, 231, -21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 21*x^19 + 231*x^18 + 77*x^17 - 3255*x^16 + 1701*x^15 + 25169*x^14 - 22890*x^13 - 120750*x^12 + 153188*x^11 + 167748*x^10 - 63896*x^9 - 667016*x^8 + 650208*x^7 - 1194704*x^6 + 2329152*x^5 + 2944032*x^4 - 4262720*x^3 - 9240448*x^2 + 9202368*x + 7854656)
 
gp: K = bnfinit(x^21 - 7*x^20 - 21*x^19 + 231*x^18 + 77*x^17 - 3255*x^16 + 1701*x^15 + 25169*x^14 - 22890*x^13 - 120750*x^12 + 153188*x^11 + 167748*x^10 - 63896*x^9 - 667016*x^8 + 650208*x^7 - 1194704*x^6 + 2329152*x^5 + 2944032*x^4 - 4262720*x^3 - 9240448*x^2 + 9202368*x + 7854656, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 21 x^{19} + 231 x^{18} + 77 x^{17} - 3255 x^{16} + 1701 x^{15} + 25169 x^{14} - 22890 x^{13} - 120750 x^{12} + 153188 x^{11} + 167748 x^{10} - 63896 x^{9} - 667016 x^{8} + 650208 x^{7} - 1194704 x^{6} + 2329152 x^{5} + 2944032 x^{4} - 4262720 x^{3} - 9240448 x^{2} + 9202368 x + 7854656 \)

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:  \(-990506497724182757810432716022494243247764430389248=-\,2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 127^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $268.15$
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:  $2, 3, 7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{4} a^{5}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} + \frac{1}{16} a^{13} + \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{18} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{32} a^{19} - \frac{1}{32} a^{18} - \frac{1}{32} a^{17} - \frac{1}{32} a^{16} + \frac{1}{32} a^{15} + \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{3}{32} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{152446932960735502413179944509090904820569485055053496721122528} a^{20} + \frac{497562774360877321010152631865742852702300658218644654068821}{76223466480367751206589972254545452410284742527526748360561264} a^{19} - \frac{33362842094273673492694282263439279217635943074368455969457}{5863343575412903938968459404195804031560364809809749873889328} a^{18} - \frac{137548186142334203352820680003499333271751734524222244860897}{76223466480367751206589972254545452410284742527526748360561264} a^{17} + \frac{1763987319888346686474673685688775978271027685715928018903651}{38111733240183875603294986127272726205142371263763374180280632} a^{16} + \frac{4301964556899931790213072148091209463165054210273095958811595}{76223466480367751206589972254545452410284742527526748360561264} a^{15} - \frac{2474538247075730699163480571313314642926726498898627798966297}{76223466480367751206589972254545452410284742527526748360561264} a^{14} - \frac{8138840664054301898183248803062988122648371773831391813576479}{76223466480367751206589972254545452410284742527526748360561264} a^{13} - \frac{1604640537596861875546790619918560021092276130963233161311389}{152446932960735502413179944509090904820569485055053496721122528} a^{12} - \frac{22858181148084250738599393090022367780812155063855911916165}{2931671787706451969484229702097902015780182404904874936944664} a^{11} - \frac{4098021444492846380103259272769803567816097497265628144419343}{38111733240183875603294986127272726205142371263763374180280632} a^{10} - \frac{87035540533182034489458608033421140417972651614270993474987}{732917946926612992371057425524475503945045601226218734236166} a^{9} - \frac{3779507391946224125318375838703077658703466361452771567771579}{19055866620091937801647493063636363102571185631881687090140316} a^{8} + \frac{7404081489144138430323888790839269683210748025149621831562403}{38111733240183875603294986127272726205142371263763374180280632} a^{7} - \frac{1682429806994242318888326876737485786977782238329172558730547}{19055866620091937801647493063636363102571185631881687090140316} a^{6} + \frac{3623493028452152996884464227279415073421361689286717958153927}{9527933310045968900823746531818181551285592815940843545070158} a^{5} + \frac{149059397270948050661268921144367379857060546837206483006431}{9527933310045968900823746531818181551285592815940843545070158} a^{4} + \frac{1431618445715359214590606641148603616838994234216809151238641}{4763966655022984450411873265909090775642796407970421772535079} a^{3} - \frac{1482436079778118432683436099012734266608109154179601928403180}{4763966655022984450411873265909090775642796407970421772535079} a^{2} + \frac{1737469642249034047003737976946340561276584655115832109424161}{4763966655022984450411873265909090775642796407970421772535079} a - \frac{1461612439609638726772715013334018850187713784640356610957357}{4763966655022984450411873265909090775642796407970421772535079}$

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:  \( 650945378943354200 \) (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

3.3.1016.1, 7.1.38423222208.5

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$