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Properties

Label 22.6.30428929951...3125.1
Degree $22$
Signature $[6, 8]$
Discriminant $5^{11}\cdot 971^{2}\cdot 25709231^{2}$
Root discriminant $19.71$
Ramified primes $5, 971, 25709231$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 10, -1, -40, -13, 37, 4, 78, 58, -108, 177, 179, -228, -47, 116, -4, 0, 0, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*x^2 + x - 1)
 
gp: K = bnfinit(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*x^2 + x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{19} - 4 x^{16} + 116 x^{15} - 47 x^{14} - 228 x^{13} + 179 x^{12} + 177 x^{11} - 108 x^{10} + 58 x^{9} + 78 x^{8} + 4 x^{7} + 37 x^{6} - 13 x^{5} - 40 x^{4} - x^{3} + 10 x^{2} + x - 1 \)

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:  \(30428929951449912919970703125=5^{11}\cdot 971^{2}\cdot 25709231^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.71$
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:  $5, 971, 25709231$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{332925175603844226134} a^{21} + \frac{16225360196109525170}{166462587801922113067} a^{20} + \frac{30779859226415276349}{332925175603844226134} a^{19} - \frac{13584446888171349539}{332925175603844226134} a^{18} - \frac{12538914129863320309}{332925175603844226134} a^{17} + \frac{36486961243901185611}{166462587801922113067} a^{16} + \frac{34689023555454607677}{166462587801922113067} a^{15} - \frac{33107646706534572169}{332925175603844226134} a^{14} + \frac{120089323238462472489}{332925175603844226134} a^{13} + \frac{16583400696064372735}{332925175603844226134} a^{12} + \frac{138186498362377521741}{332925175603844226134} a^{11} + \frac{67234274046828626551}{332925175603844226134} a^{10} + \frac{2790939407421424255}{166462587801922113067} a^{9} - \frac{10014806344577005144}{166462587801922113067} a^{8} + \frac{98362305985898273499}{332925175603844226134} a^{7} + \frac{80179321063684944511}{166462587801922113067} a^{6} + \frac{82766648862762730125}{166462587801922113067} a^{5} + \frac{9776779135101216524}{166462587801922113067} a^{4} - \frac{42990118571195576073}{166462587801922113067} a^{3} + \frac{68983914236788723181}{332925175603844226134} a^{2} - \frac{20099626289428828628}{166462587801922113067} a + \frac{26025463368267896706}{166462587801922113067}$

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:  \( 376545.621481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 11.3.24963663301.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ $22$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $18{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
971Data not computed
25709231Data not computed