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Properties

Label 22.0.61415850852...7824.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{68}\cdot 3^{20}\cdot 17\cdot 97\cdot 337^{8}\cdot 8297\cdot 11273\cdot 26921\cdot 310501^{8}$
Root discriminant $97{,}808.45$
Ramified primes $2, 3, 17, 97, 337, 8297, 11273, 26921, 310501$
Class number Not computed
Class group Not computed
Galois group 22T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![597908782748023056, 0, 522455049623219279, 0, 96794448620299288, 0, 8187635841570693, 0, 397142176666080, 0, 12216440211264, 0, 249908957112, 0, 3455500056, 0, 31982160, 0, 190059, 0, 656, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 656*x^20 + 190059*x^18 + 31982160*x^16 + 3455500056*x^14 + 249908957112*x^12 + 12216440211264*x^10 + 397142176666080*x^8 + 8187635841570693*x^6 + 96794448620299288*x^4 + 522455049623219279*x^2 + 597908782748023056)
 
gp: K = bnfinit(x^22 + 656*x^20 + 190059*x^18 + 31982160*x^16 + 3455500056*x^14 + 249908957112*x^12 + 12216440211264*x^10 + 397142176666080*x^8 + 8187635841570693*x^6 + 96794448620299288*x^4 + 522455049623219279*x^2 + 597908782748023056, 1)
 

Normalized defining polynomial

\( x^{22} + 656 x^{20} + 190059 x^{18} + 31982160 x^{16} + 3455500056 x^{14} + 249908957112 x^{12} + 12216440211264 x^{10} + 397142176666080 x^{8} + 8187635841570693 x^{6} + 96794448620299288 x^{4} + 522455049623219279 x^{2} + 597908782748023056 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-61415850852267200397515695951218522185201753278017240901376750286638166845798793940070572804439992043594317824=-\,2^{68}\cdot 3^{20}\cdot 17\cdot 97\cdot 337^{8}\cdot 8297\cdot 11273\cdot 26921\cdot 310501^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97{,}808.45$
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:  $2, 3, 17, 97, 337, 8297, 11273, 26921, 310501$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718490685}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{285314373863498156819074950717143524387487689154714929482}{1315620922890192728084676057556238674784696872714445759239} a^{16} + \frac{5944282035118542332199914159684414566100617890003123549920}{14471830151792120008931436633118625422631665599858903351629} a^{14} + \frac{5079880002412714118012206989965807874620296635018234819967}{14471830151792120008931436633118625422631665599858903351629} a^{12} - \frac{5736856693441053286414799276111616633163315487318148987591}{14471830151792120008931436633118625422631665599858903351629} a^{10} - \frac{2238730398598376248383505132029243922465089711141898142967}{14471830151792120008931436633118625422631665599858903351629} a^{8} - \frac{4752503954843329533704203467523599594219342664324526636535}{14471830151792120008931436633118625422631665599858903351629} a^{6} + \frac{25924383890974682371741859663195175816388224369793171662225}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{3883945488032126807101248456736692867503436736718451565533}{57887320607168480035725746532474501690526662399435613406516} a^{2} + \frac{6596098091164470811536283423510131792881603828929434631956}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{347323923643010880214354479194847010143159974396613680439096} a^{21} + \frac{20232524059240298510684950852616176915805712411409995782101}{173661961821505440107177239597423505071579987198306840219548} a^{19} - \frac{1696040088041523603843442658512430040634680458254065665215}{10524967383121541824677408460449909398277574981715566073912} a^{17} - \frac{1}{2} a^{16} + \frac{5814657056450463725010464570986944235227324848286821708863}{14471830151792120008931436633118625422631665599858903351629} a^{15} + \frac{3258618359034139021157273937180738882875327039146189695266}{14471830151792120008931436633118625422631665599858903351629} a^{13} + \frac{6279772293655884456729918437207376605788613552043093511216}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{6862793342796330629735134127887772057571651181405801985320}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{6443831083422171748848351071972046112279275689208697236392}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{49245859310176919245145126644742776418397254276171222852441}{115774641214336960071451493064949003381053324798871226813032} a^{5} + \frac{77653093090840596645572277253784093390722442430864742651193}{173661961821505440107177239597423505071579987198306840219548} a^{3} + \frac{40856222516450003255076570327159152594158080915576641879453}{347323923643010880214354479194847010143159974396613680439096} a - \frac{1}{2}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16220160
The 104 conjugacy class representatives for t22n44 are not computed
Character table for t22n44 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 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 $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $22$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
17Data not computed
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
97.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
97.10.0.1$x^{10} - 2 x + 57$$1$$10$$0$$C_{10}$$[\ ]^{10}$
337Data not computed
8297Data not computed
11273Data not computed
26921Data not computed
310501Data not computed