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Properties

Label 22.22.4598055053...1437.1
Degree $22$
Signature $[22, 0]$
Discriminant $19^{11}\cdot 23^{21}$
Root discriminant $86.94$
Ramified primes $19, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![194249261, -1317296136, 1317296136, 3624110114, -3624110114, -2799718011, 2799718011, 1054578864, -1054578864, -230186761, 230186761, 31438239, -31438239, -2774261, 2774261, 158239, -158239, -5636, 5636, 114, -114, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 114*x^20 + 114*x^19 + 5636*x^18 - 5636*x^17 - 158239*x^16 + 158239*x^15 + 2774261*x^14 - 2774261*x^13 - 31438239*x^12 + 31438239*x^11 + 230186761*x^10 - 230186761*x^9 - 1054578864*x^8 + 1054578864*x^7 + 2799718011*x^6 - 2799718011*x^5 - 3624110114*x^4 + 3624110114*x^3 + 1317296136*x^2 - 1317296136*x + 194249261)
 
gp: K = bnfinit(x^22 - x^21 - 114*x^20 + 114*x^19 + 5636*x^18 - 5636*x^17 - 158239*x^16 + 158239*x^15 + 2774261*x^14 - 2774261*x^13 - 31438239*x^12 + 31438239*x^11 + 230186761*x^10 - 230186761*x^9 - 1054578864*x^8 + 1054578864*x^7 + 2799718011*x^6 - 2799718011*x^5 - 3624110114*x^4 + 3624110114*x^3 + 1317296136*x^2 - 1317296136*x + 194249261, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 114 x^{20} + 114 x^{19} + 5636 x^{18} - 5636 x^{17} - 158239 x^{16} + 158239 x^{15} + 2774261 x^{14} - 2774261 x^{13} - 31438239 x^{12} + 31438239 x^{11} + 230186761 x^{10} - 230186761 x^{9} - 1054578864 x^{8} + 1054578864 x^{7} + 2799718011 x^{6} - 2799718011 x^{5} - 3624110114 x^{4} + 3624110114 x^{3} + 1317296136 x^{2} - 1317296136 x + 194249261 \)

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:  \(4598055053342647107748042243736831875581437=19^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.94$
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:  $19, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(437=19\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(132,·)$, $\chi_{437}(113,·)$, $\chi_{437}(77,·)$, $\chi_{437}(398,·)$, $\chi_{437}(400,·)$, $\chi_{437}(248,·)$, $\chi_{437}(210,·)$, $\chi_{437}(341,·)$, $\chi_{437}(324,·)$, $\chi_{437}(96,·)$, $\chi_{437}(227,·)$, $\chi_{437}(37,·)$, $\chi_{437}(39,·)$, $\chi_{437}(360,·)$, $\chi_{437}(381,·)$, $\chi_{437}(305,·)$, $\chi_{437}(436,·)$, $\chi_{437}(56,·)$, $\chi_{437}(58,·)$, $\chi_{437}(379,·)$, $\chi_{437}(189,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{22885229} a^{12} - \frac{5895364}{22885229} a^{11} - \frac{60}{22885229} a^{10} + \frac{3851814}{22885229} a^{9} + \frac{1350}{22885229} a^{8} - \frac{8380593}{22885229} a^{7} - \frac{14000}{22885229} a^{6} + \frac{10395809}{22885229} a^{5} + \frac{65625}{22885229} a^{4} - \frac{4434705}{22885229} a^{3} - \frac{112500}{22885229} a^{2} + \frac{4434705}{22885229} a + \frac{31250}{22885229}$, $\frac{1}{22885229} a^{13} - \frac{65}{22885229} a^{11} - \frac{6591591}{22885229} a^{10} + \frac{1625}{22885229} a^{9} + \frac{9186344}{22885229} a^{8} - \frac{19500}{22885229} a^{7} - \frac{564417}{22885229} a^{6} + \frac{113750}{22885229} a^{5} + \frac{4031550}{22885229} a^{4} - \frac{284375}{22885229} a^{3} - \frac{10078875}{22885229} a^{2} + \frac{203125}{22885229} a + \frac{4031550}{22885229}$, $\frac{1}{22885229} a^{14} - \frac{741358}{22885229} a^{11} - \frac{2275}{22885229} a^{10} + \frac{7816735}{22885229} a^{9} + \frac{68250}{22885229} a^{8} + \frac{3942534}{22885229} a^{7} - \frac{796250}{22885229} a^{6} - \frac{6797735}{22885229} a^{5} + \frac{3981250}{22885229} a^{4} - \frac{826723}{22885229} a^{3} - \frac{7109375}{22885229} a^{2} - \frac{5220602}{22885229} a + \frac{2031250}{22885229}$, $\frac{1}{22885229} a^{15} - \frac{2625}{22885229} a^{11} + \frac{9105713}{22885229} a^{10} + \frac{87500}{22885229} a^{9} - \frac{2174242}{22885229} a^{8} - \frac{1181250}{22885229} a^{7} + \frac{4084231}{22885229} a^{6} + \frac{7350000}{22885229} a^{5} - \frac{3204827}{22885229} a^{4} + \frac{3744604}{22885229} a^{3} + \frac{8664103}{22885229} a^{2} - \frac{8822729}{22885229} a + \frac{7585752}{22885229}$, $\frac{1}{22885229} a^{16} + \frac{4190017}{22885229} a^{11} - \frac{70000}{22885229} a^{10} - \frac{6433710}{22885229} a^{9} + \frac{2362500}{22885229} a^{8} - \frac{2267325}{22885229} a^{7} - \frac{6514771}{22885229} a^{6} + \frac{6600830}{22885229} a^{5} - \frac{7071603}{22885229} a^{4} - \frac{6740190}{22885229} a^{3} - \frac{6627252}{22885229} a^{2} + \frac{104816}{22885229} a - \frac{9509666}{22885229}$, $\frac{1}{22885229} a^{17} - \frac{85000}{22885229} a^{11} - \frac{6770209}{22885229} a^{10} + \frac{3187500}{22885229} a^{9} - \frac{6138712}{22885229} a^{8} - \frac{129542}{22885229} a^{7} - \frac{10888326}{22885229} a^{6} - \frac{7977}{22885229} a^{5} - \frac{10579380}{22885229} a^{4} + \frac{4108015}{22885229} a^{3} + \frac{9955603}{22885229} a^{2} + \frac{2640296}{22885229} a + \frac{11249088}{22885229}$, $\frac{1}{22885229} a^{18} + \frac{5149204}{22885229} a^{11} - \frac{1912500}{22885229} a^{10} + \frac{1965214}{22885229} a^{9} + \frac{194313}{22885229} a^{8} + \frac{10114986}{22885229} a^{7} + \frac{23931}{22885229} a^{6} - \frac{11276528}{22885229} a^{5} - \frac{1762861}{22885229} a^{4} + \frac{2637462}{22885229} a^{3} + \frac{6166018}{22885229} a^{2} - \frac{4318000}{22885229} a + \frac{1563436}{22885229}$, $\frac{1}{22885229} a^{19} - \frac{2422500}{22885229} a^{11} - \frac{9475752}{22885229} a^{10} + \frac{5359084}{22885229} a^{9} - \frac{7086027}{22885229} a^{8} + \frac{11154656}{22885229} a^{7} - \frac{10891878}{22885229} a^{6} + \frac{9548133}{22885229} a^{5} + \frac{9416376}{22885229} a^{4} + \frac{5001432}{22885229} a^{3} + \frac{10215552}{22885229} a^{2} + \frac{2728022}{22885229} a - \frac{6579901}{22885229}$, $\frac{1}{22885229} a^{20} - \frac{1608302}{22885229} a^{11} - \frac{2679542}{22885229} a^{10} - \frac{4976426}{22885229} a^{9} + \frac{8941909}{22885229} a^{8} - \frac{7313440}{22885229} a^{7} + \frac{10457511}{22885229} a^{6} - \frac{10452342}{22885229} a^{5} - \frac{2121931}{22885229} a^{4} - \frac{3827020}{22885229} a^{3} - \frac{11215046}{22885229} a^{2} + \frac{7462671}{22885229} a - \frac{1212532}{22885229}$, $\frac{1}{22885229} a^{21} + \frac{5065062}{22885229} a^{11} - \frac{9933630}{22885229} a^{10} - \frac{5077189}{22885229} a^{9} - \frac{10202495}{22885229} a^{8} - \frac{6668506}{22885229} a^{7} - \frac{7615006}{22885229} a^{6} - \frac{6744578}{22885229} a^{5} - \frac{5684418}{22885229} a^{4} + \frac{8563726}{22885229} a^{3} + \frac{4108145}{22885229} a^{2} + \frac{1893925}{22885229} a + \frac{3474616}{22885229}$

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:  \( 205240201072000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{437}) \), \(\Q(\zeta_{23})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22$ $22$ $22$ $22$ $22$ $22$ $22$ R R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
19Data not computed
23Data not computed