↑

Properties

Label 22.22.2681925502...9152.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{33}\cdot 3^{20}\cdot 11^{23}$
Root discriminant $94.19$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{11}$ (as 22T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1650137, 6107310, 85745, -21398190, -12274108, 32367456, 24186393, -27704688, -22009119, 14632332, 11274747, -4810374, -3450678, 940962, 642477, -101112, -70653, 5016, 4180, -66, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 110*x^20 - 66*x^19 + 4180*x^18 + 5016*x^17 - 70653*x^16 - 101112*x^15 + 642477*x^14 + 940962*x^13 - 3450678*x^12 - 4810374*x^11 + 11274747*x^10 + 14632332*x^9 - 22009119*x^8 - 27704688*x^7 + 24186393*x^6 + 32367456*x^5 - 12274108*x^4 - 21398190*x^3 + 85745*x^2 + 6107310*x + 1650137)
 
gp: K = bnfinit(x^22 - 110*x^20 - 66*x^19 + 4180*x^18 + 5016*x^17 - 70653*x^16 - 101112*x^15 + 642477*x^14 + 940962*x^13 - 3450678*x^12 - 4810374*x^11 + 11274747*x^10 + 14632332*x^9 - 22009119*x^8 - 27704688*x^7 + 24186393*x^6 + 32367456*x^5 - 12274108*x^4 - 21398190*x^3 + 85745*x^2 + 6107310*x + 1650137, 1)
 

Normalized defining polynomial

\( x^{22} - 110 x^{20} - 66 x^{19} + 4180 x^{18} + 5016 x^{17} - 70653 x^{16} - 101112 x^{15} + 642477 x^{14} + 940962 x^{13} - 3450678 x^{12} - 4810374 x^{11} + 11274747 x^{10} + 14632332 x^{9} - 22009119 x^{8} - 27704688 x^{7} + 24186393 x^{6} + 32367456 x^{5} - 12274108 x^{4} - 21398190 x^{3} + 85745 x^{2} + 6107310 x + 1650137 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26819255020464372185020972278794137897009152=2^{33}\cdot 3^{20}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23} a^{20} + \frac{1}{23} a^{19} + \frac{11}{23} a^{18} - \frac{4}{23} a^{17} - \frac{1}{23} a^{16} + \frac{4}{23} a^{15} + \frac{2}{23} a^{14} - \frac{5}{23} a^{13} + \frac{7}{23} a^{11} - \frac{4}{23} a^{10} - \frac{8}{23} a^{9} + \frac{4}{23} a^{8} - \frac{5}{23} a^{7} + \frac{10}{23} a^{6} - \frac{7}{23} a^{5} + \frac{4}{23} a^{4} + \frac{8}{23} a^{3} + \frac{8}{23} a^{2} - \frac{5}{23}$, $\frac{1}{117287396327961242918310353946308293166260018291571} a^{21} + \frac{1115467766414827885644670094121395167184630921065}{117287396327961242918310353946308293166260018291571} a^{20} - \frac{48287407052657365104011114300446518610877566889013}{117287396327961242918310353946308293166260018291571} a^{19} + \frac{2942245198405376448886347725122291668181196839419}{6899258607527131936371197290959311362721177546563} a^{18} + \frac{2427516178878281122158178976652125046827077681496}{5099452014259184474709145823752534485489566012677} a^{17} - \frac{9206112465113092006233050419753370276819626308159}{117287396327961242918310353946308293166260018291571} a^{16} - \frac{40445896949228541967740782326790280123637218005900}{117287396327961242918310353946308293166260018291571} a^{15} + \frac{7658712298138522166462450320103727751872729677212}{117287396327961242918310353946308293166260018291571} a^{14} - \frac{38278193947225314441535349842572849945229467044454}{117287396327961242918310353946308293166260018291571} a^{13} + \frac{22896857098622184337414031258598348737769558658943}{117287396327961242918310353946308293166260018291571} a^{12} - \frac{641186097881649208043842126446870923541421161567}{5099452014259184474709145823752534485489566012677} a^{11} + \frac{14458489625241485768710219906186877672657564241691}{117287396327961242918310353946308293166260018291571} a^{10} - \frac{2774895171352945254430272826046998573076259241320}{117287396327961242918310353946308293166260018291571} a^{9} + \frac{54477420180168628436032998664294066213241834451086}{117287396327961242918310353946308293166260018291571} a^{8} + \frac{49094569713746196859839534547694033151008911175327}{117287396327961242918310353946308293166260018291571} a^{7} + \frac{14388187344220928725001322073234561208345707186248}{117287396327961242918310353946308293166260018291571} a^{6} - \frac{548532961589081457111888914082466429322515069782}{5099452014259184474709145823752534485489566012677} a^{5} - \frac{45459831324393067828341678593464116142569428432571}{117287396327961242918310353946308293166260018291571} a^{4} + \frac{858898792259984862406597280236174701049022388771}{6899258607527131936371197290959311362721177546563} a^{3} + \frac{41597882881529653408602319461346517307913115977343}{117287396327961242918310353946308293166260018291571} a^{2} + \frac{12833313214937225054805543794936218180027654309360}{117287396327961242918310353946308293166260018291571} a + \frac{7136560943965696478389381106282502617420913400985}{117287396327961242918310353946308293166260018291571}$

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:  $21$
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:  \( 3109756315010000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{11}$ (as 22T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 11 conjugacy class representatives for $F_{11}$
Character table for $F_{11}$

Intermediate fields

\(\Q(\sqrt{22}) \), 11.11.552054582080600113152.1

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

Sibling fields

Degree 11 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
11Data not computed