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Properties

Label 22.10.1211544925...8125.1
Degree $22$
Signature $[10, 6]$
Discriminant $5^{11}\cdot 19457^{2}\cdot 8095783^{2}$
Root discriminant $23.30$
Ramified primes $5, 19457, 8095783$
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, 2, 4, -12, -4, 16, 27, 45, 32, 64, -56, -359, -301, 64, 178, 54, 10, 7, -6, -1, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 4*x^20 - x^19 - 6*x^18 + 7*x^17 + 10*x^16 + 54*x^15 + 178*x^14 + 64*x^13 - 301*x^12 - 359*x^11 - 56*x^10 + 64*x^9 + 32*x^8 + 45*x^7 + 27*x^6 + 16*x^5 - 4*x^4 - 12*x^3 + 4*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 4*x^20 - x^19 - 6*x^18 + 7*x^17 + 10*x^16 + 54*x^15 + 178*x^14 + 64*x^13 - 301*x^12 - 359*x^11 - 56*x^10 + 64*x^9 + 32*x^8 + 45*x^7 + 27*x^6 + 16*x^5 - 4*x^4 - 12*x^3 + 4*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 4 x^{20} - x^{19} - 6 x^{18} + 7 x^{17} + 10 x^{16} + 54 x^{15} + 178 x^{14} + 64 x^{13} - 301 x^{12} - 359 x^{11} - 56 x^{10} + 64 x^{9} + 32 x^{8} + 45 x^{7} + 27 x^{6} + 16 x^{5} - 4 x^{4} - 12 x^{3} + 4 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1211544925921916910574267578125=5^{11}\cdot 19457^{2}\cdot 8095783^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.30$
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:  $5, 19457, 8095783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{484865932711622746039} a^{21} - \frac{139400163043836945619}{484865932711622746039} a^{20} + \frac{39399409934115060370}{484865932711622746039} a^{19} + \frac{141032539692092566565}{484865932711622746039} a^{18} - \frac{55163495596007235463}{484865932711622746039} a^{17} - \frac{87107510236525136751}{484865932711622746039} a^{16} - \frac{26075899950271865446}{69266561815946106577} a^{15} + \frac{188848729007096802446}{484865932711622746039} a^{14} - \frac{205735968475227914958}{484865932711622746039} a^{13} - \frac{84173371447117843094}{484865932711622746039} a^{12} - \frac{225425861321974968742}{484865932711622746039} a^{11} - \frac{122345624094058541957}{484865932711622746039} a^{10} + \frac{98504131138551917151}{484865932711622746039} a^{9} - \frac{32562838690612442944}{69266561815946106577} a^{8} + \frac{94453054428458800631}{484865932711622746039} a^{7} - \frac{22011832904697092512}{484865932711622746039} a^{6} - \frac{137136192653031052794}{484865932711622746039} a^{5} - \frac{233138125285224754692}{484865932711622746039} a^{4} - \frac{126609621810010556904}{484865932711622746039} a^{3} + \frac{19140101447455433325}{484865932711622746039} a^{2} + \frac{9312748524937186736}{484865932711622746039} a - \frac{46481079985341878830}{484865932711622746039}$

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:  $15$
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:  \( 5475014.69169 \) (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

\(\Q(\sqrt{5}) \), 11.5.157519649831.1

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 $22$ $22$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $18{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
19457Data not computed
8095783Data not computed