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Properties

Label 22.6.26612486644...3312.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{24}\cdot 3^{11}\cdot 11^{23}$
Root discriminant $45.26$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T14

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9888, 3696, 32560, 72512, -37840, -400400, 704264, -515240, 172414, 22077, -90409, 98910, -65956, 23056, -968, -1540, -550, 825, -231, -44, 44, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 44*x^20 - 44*x^19 - 231*x^18 + 825*x^17 - 550*x^16 - 1540*x^15 - 968*x^14 + 23056*x^13 - 65956*x^12 + 98910*x^11 - 90409*x^10 + 22077*x^9 + 172414*x^8 - 515240*x^7 + 704264*x^6 - 400400*x^5 - 37840*x^4 + 72512*x^3 + 32560*x^2 + 3696*x - 9888)
 
gp: K = bnfinit(x^22 - 11*x^21 + 44*x^20 - 44*x^19 - 231*x^18 + 825*x^17 - 550*x^16 - 1540*x^15 - 968*x^14 + 23056*x^13 - 65956*x^12 + 98910*x^11 - 90409*x^10 + 22077*x^9 + 172414*x^8 - 515240*x^7 + 704264*x^6 - 400400*x^5 - 37840*x^4 + 72512*x^3 + 32560*x^2 + 3696*x - 9888, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 44 x^{20} - 44 x^{19} - 231 x^{18} + 825 x^{17} - 550 x^{16} - 1540 x^{15} - 968 x^{14} + 23056 x^{13} - 65956 x^{12} + 98910 x^{11} - 90409 x^{10} + 22077 x^{9} + 172414 x^{8} - 515240 x^{7} + 704264 x^{6} - 400400 x^{5} - 37840 x^{4} + 72512 x^{3} + 32560 x^{2} + 3696 x - 9888 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2661248664423333685102326194280333312=2^{24}\cdot 3^{11}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.26$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{18} - \frac{1}{8} a^{17} + \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{5}{16} a^{7} + \frac{3}{16} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{20} - \frac{1}{32} a^{19} - \frac{1}{16} a^{17} - \frac{3}{32} a^{16} - \frac{1}{32} a^{15} + \frac{3}{16} a^{14} + \frac{1}{16} a^{13} - \frac{1}{4} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{11}{32} a^{8} + \frac{15}{32} a^{7} + \frac{7}{16} a^{6} + \frac{7}{16} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{280020657167251580015667237342099955541564749177760} a^{21} - \frac{480212776090060817642474007913493572916729867519}{35002582145906447501958404667762494442695593647220} a^{20} - \frac{7286551491179744056817639311227135341057985343339}{280020657167251580015667237342099955541564749177760} a^{19} - \frac{45728295747898829886611460591459307911772248588}{1750129107295322375097920233388124722134779682361} a^{18} - \frac{9137836420447236740840755586862416403135439311321}{280020657167251580015667237342099955541564749177760} a^{17} - \frac{5432076264236533692775531169807307005516045060921}{70005164291812895003916809335524988885391187294440} a^{16} + \frac{25811502164406400067351552703280336168556125281819}{280020657167251580015667237342099955541564749177760} a^{15} - \frac{30842420962699390063837827200570284514388364371467}{140010328583625790007833618671049977770782374588880} a^{14} + \frac{33672792126273623660225415452349959233614965145773}{140010328583625790007833618671049977770782374588880} a^{13} + \frac{2264312523893865508263224401271917592449499496269}{14001032858362579000783361867104997777078237458888} a^{12} - \frac{2456354239976061494293949158237821201181873738071}{17501291072953223750979202333881247221347796823610} a^{11} + \frac{25102324487168124333027857941073901895550854354853}{140010328583625790007833618671049977770782374588880} a^{10} + \frac{2004742272976549386119512247438779099676731487613}{56004131433450316003133447468419991108312949835552} a^{9} - \frac{21272540743233585130254450965162430432352178809269}{140010328583625790007833618671049977770782374588880} a^{8} + \frac{19559473879962385120822958529014453138139839306927}{280020657167251580015667237342099955541564749177760} a^{7} + \frac{27442198357877286492380669750587928239300174977219}{140010328583625790007833618671049977770782374588880} a^{6} + \frac{650323487441593214877769468546664775411783349591}{10770025275663522308294893743926921366983259583760} a^{5} + \frac{708510762994883417427655320015923658970742411087}{5385012637831761154147446871963460683491629791880} a^{4} - \frac{12469022278062730830144326054393356304819738825841}{70005164291812895003916809335524988885391187294440} a^{3} - \frac{15612279801624093833627934972211087825855334844923}{35002582145906447501958404667762494442695593647220} a^{2} + \frac{10176283039135265672156409843805579877578883293613}{35002582145906447501958404667762494442695593647220} a - \frac{7402252283634789907396576089705805801076002986773}{17501291072953223750979202333881247221347796823610}$

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:  $13$
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:  \( 158674344921 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T14:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 13 conjugacy class representatives for t22n14
Character table for t22n14

Intermediate fields

\(\Q(\sqrt{33}) \)

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed