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Properties

Label 22.10.4098058278...8729.1
Degree $22$
Signature $[10, 6]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 40, 565, 3280, 6218, -2532, -11948, 3976, 9862, -3466, -5194, -93, 2996, -1094, -934, 429, 255, -127, -66, 51, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 66*x^18 - 127*x^17 + 255*x^16 + 429*x^15 - 934*x^14 - 1094*x^13 + 2996*x^12 - 93*x^11 - 5194*x^10 - 3466*x^9 + 9862*x^8 + 3976*x^7 - 11948*x^6 - 2532*x^5 + 6218*x^4 + 3280*x^3 + 565*x^2 + 40*x + 1)
 
gp: K = bnfinit(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 66*x^18 - 127*x^17 + 255*x^16 + 429*x^15 - 934*x^14 - 1094*x^13 + 2996*x^12 - 93*x^11 - 5194*x^10 - 3466*x^9 + 9862*x^8 + 3976*x^7 - 11948*x^6 - 2532*x^5 + 6218*x^4 + 3280*x^3 + 565*x^2 + 40*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 7 x^{20} + 51 x^{19} - 66 x^{18} - 127 x^{17} + 255 x^{16} + 429 x^{15} - 934 x^{14} - 1094 x^{13} + 2996 x^{12} - 93 x^{11} - 5194 x^{10} - 3466 x^{9} + 9862 x^{8} + 3976 x^{7} - 11948 x^{6} - 2532 x^{5} + 6218 x^{4} + 3280 x^{3} + 565 x^{2} + 40 x + 1 \)

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:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{15}{47} a^{19} - \frac{18}{47} a^{18} + \frac{22}{47} a^{17} - \frac{7}{47} a^{16} - \frac{21}{47} a^{15} - \frac{21}{47} a^{14} - \frac{15}{47} a^{13} + \frac{13}{47} a^{12} - \frac{7}{47} a^{11} - \frac{14}{47} a^{10} - \frac{23}{47} a^{9} + \frac{14}{47} a^{8} + \frac{5}{47} a^{7} + \frac{11}{47} a^{6} + \frac{14}{47} a^{5} + \frac{15}{47} a^{4} + \frac{9}{47} a^{3} + \frac{1}{47} a^{2} - \frac{7}{47} a - \frac{4}{47}$, $\frac{1}{415166355281277487264805871439941702200173} a^{21} - \frac{1918962207019687390580404890923189217946}{415166355281277487264805871439941702200173} a^{20} + \frac{89229871152463235357723253048351759837280}{415166355281277487264805871439941702200173} a^{19} - \frac{141514691199407824027343378679641353845741}{415166355281277487264805871439941702200173} a^{18} + \frac{117716466511233766898589411396343905907418}{415166355281277487264805871439941702200173} a^{17} - \frac{62217983925429012680786722786637266708836}{415166355281277487264805871439941702200173} a^{16} - \frac{93696678530095416519275674651468963607374}{415166355281277487264805871439941702200173} a^{15} - \frac{68763871650273147737529635296076957342660}{415166355281277487264805871439941702200173} a^{14} + \frac{2676640066820240624844844488274661991416}{415166355281277487264805871439941702200173} a^{13} - \frac{61611636996547495542492268603246098355622}{415166355281277487264805871439941702200173} a^{12} + \frac{194898258359018559705412625622109840020741}{415166355281277487264805871439941702200173} a^{11} - \frac{96091150996488188681920536662069207714211}{415166355281277487264805871439941702200173} a^{10} + \frac{118673229247784223325946940453238803821418}{415166355281277487264805871439941702200173} a^{9} + \frac{36667328093052807914595692506228202490576}{415166355281277487264805871439941702200173} a^{8} - \frac{44737726210235852000301813774020742993975}{415166355281277487264805871439941702200173} a^{7} + \frac{63894079562352864342694778807057609447122}{415166355281277487264805871439941702200173} a^{6} - \frac{99038726675777641117427861151536918656887}{415166355281277487264805871439941702200173} a^{5} + \frac{118867741435795592851547725726034662257652}{415166355281277487264805871439941702200173} a^{4} + \frac{45475534617674660286144138605418256958231}{415166355281277487264805871439941702200173} a^{3} + \frac{71074430936174050709945135568535651063259}{415166355281277487264805871439941702200173} a^{2} - \frac{41093276989486732009797474423740683755745}{415166355281277487264805871439941702200173} a + \frac{139342198942165619831858973846889346867540}{415166355281277487264805871439941702200173}$

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:  \( 345465500.198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
47Data not computed