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Properties

Label 22.22.4807060044...8125.2
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 74843^{8}$
Root discriminant $132.40$
Ramified primes $5, 74843$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T13

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![247031, -639053, -2533933, 5211574, 7359840, -15249644, -7618260, 19735158, 2499505, -13086344, 754889, 4759643, -758660, -975861, 212871, 114096, -29089, -7414, 2085, 243, -74, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 74*x^20 + 243*x^19 + 2085*x^18 - 7414*x^17 - 29089*x^16 + 114096*x^15 + 212871*x^14 - 975861*x^13 - 758660*x^12 + 4759643*x^11 + 754889*x^10 - 13086344*x^9 + 2499505*x^8 + 19735158*x^7 - 7618260*x^6 - 15249644*x^5 + 7359840*x^4 + 5211574*x^3 - 2533933*x^2 - 639053*x + 247031)
 
gp: K = bnfinit(x^22 - 3*x^21 - 74*x^20 + 243*x^19 + 2085*x^18 - 7414*x^17 - 29089*x^16 + 114096*x^15 + 212871*x^14 - 975861*x^13 - 758660*x^12 + 4759643*x^11 + 754889*x^10 - 13086344*x^9 + 2499505*x^8 + 19735158*x^7 - 7618260*x^6 - 15249644*x^5 + 7359840*x^4 + 5211574*x^3 - 2533933*x^2 - 639053*x + 247031, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 74 x^{20} + 243 x^{19} + 2085 x^{18} - 7414 x^{17} - 29089 x^{16} + 114096 x^{15} + 212871 x^{14} - 975861 x^{13} - 758660 x^{12} + 4759643 x^{11} + 754889 x^{10} - 13086344 x^{9} + 2499505 x^{8} + 19735158 x^{7} - 7618260 x^{6} - 15249644 x^{5} + 7359840 x^{4} + 5211574 x^{3} - 2533933 x^{2} - 639053 x + 247031 \)

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:  \(48070600443250243342242248871561405585986328125=5^{11}\cdot 74843^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.40$
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:  $5, 74843$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} + \frac{1}{3} a^{15} + \frac{4}{9} a^{14} + \frac{2}{9} a^{13} - \frac{4}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{19} - \frac{1}{9} a^{17} + \frac{1}{9} a^{15} + \frac{4}{9} a^{14} - \frac{1}{3} a^{13} + \frac{4}{9} a^{12} - \frac{4}{9} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{97815141} a^{20} + \frac{1477160}{97815141} a^{19} + \frac{604730}{32605047} a^{18} + \frac{5165282}{97815141} a^{17} - \frac{9608375}{97815141} a^{16} + \frac{1392474}{3622783} a^{15} - \frac{12809527}{32605047} a^{14} + \frac{13240325}{32605047} a^{13} + \frac{5506342}{32605047} a^{12} + \frac{4829029}{32605047} a^{11} + \frac{22842157}{97815141} a^{10} + \frac{20308351}{97815141} a^{9} + \frac{38776022}{97815141} a^{8} - \frac{4640933}{10868349} a^{7} - \frac{15705967}{32605047} a^{6} + \frac{11317754}{32605047} a^{5} - \frac{421387}{32605047} a^{4} - \frac{8815256}{97815141} a^{3} - \frac{453070}{97815141} a^{2} + \frac{4998548}{32605047} a + \frac{16266484}{97815141}$, $\frac{1}{564699346274368916136023871038691170541437760418551} a^{21} + \frac{700991272988451792458609603620238115996692}{564699346274368916136023871038691170541437760418551} a^{20} + \frac{1523736643599754507398163006841962129140974192578}{188233115424789638712007957012897056847145920139517} a^{19} + \frac{2514810001507458186434552855000980050268716335885}{564699346274368916136023871038691170541437760418551} a^{18} - \frac{2209893060373356754734120050089129202931650782781}{564699346274368916136023871038691170541437760418551} a^{17} - \frac{20972019439780518994235241009244386166364077753592}{188233115424789638712007957012897056847145920139517} a^{16} + \frac{16703859299119379737507507070717347368696440265107}{188233115424789638712007957012897056847145920139517} a^{15} + \frac{58907237906518504154837207682091797878346900315899}{188233115424789638712007957012897056847145920139517} a^{14} - \frac{23041481560175787875599610950547507522108853615950}{188233115424789638712007957012897056847145920139517} a^{13} + \frac{900873541297633079117724039569476389280389942964}{188233115424789638712007957012897056847145920139517} a^{12} - \frac{203296599726915464452563830243757844868645499227684}{564699346274368916136023871038691170541437760418551} a^{11} + \frac{217024113778303121602232477572883020455225433740067}{564699346274368916136023871038691170541437760418551} a^{10} + \frac{278746679806176152779544157101178971881100718290753}{564699346274368916136023871038691170541437760418551} a^{9} + \frac{35371380150499342622272335809998962248511331258253}{188233115424789638712007957012897056847145920139517} a^{8} - \frac{76164044583898616688058495151078252017034868520489}{188233115424789638712007957012897056847145920139517} a^{7} - \frac{80832347319168814935479885889813770300526496206097}{188233115424789638712007957012897056847145920139517} a^{6} + \frac{40603552508487701726232544328956436352215814298334}{188233115424789638712007957012897056847145920139517} a^{5} - \frac{242574773907820598278527247040945443168646426130398}{564699346274368916136023871038691170541437760418551} a^{4} + \frac{170203843270696183817100234370262661496033222185233}{564699346274368916136023871038691170541437760418551} a^{3} - \frac{51239997358322189325282968383911920712099234616136}{188233115424789638712007957012897056847145920139517} a^{2} + \frac{234493634402764065928271779517130248129379864482638}{564699346274368916136023871038691170541437760418551} a - \frac{48523383152337779092272968709072397906573046962364}{188233115424789638712007957012897056847145920139517}$

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:  \( 23316714717000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T13:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 16 conjugacy class representatives for t22n13
Character table for t22n13

Intermediate fields

\(\Q(\sqrt{5}) \), 11.11.31376518243389673201.1

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

Sibling fields

Degree 24 sibling: data not computed
Arithmetically equvalently 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 $22$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
74843Data not computed