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Properties

Label 22.0.19273234433...6875.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,5^{11}\cdot 23^{21}$
Root discriminant $44.60$
Ramified primes $5, 23$
Class number $662$ (GRH)
Class group $[662]$ (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![64079, -64056, 64056, -63550, 63550, -60261, 60261, -50394, 50394, -33949, 33949, -17205, 17205, -6257, 6257, -1565, 1565, -254, 254, -24, 24, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 24*x^20 - 24*x^19 + 254*x^18 - 254*x^17 + 1565*x^16 - 1565*x^15 + 6257*x^14 - 6257*x^13 + 17205*x^12 - 17205*x^11 + 33949*x^10 - 33949*x^9 + 50394*x^8 - 50394*x^7 + 60261*x^6 - 60261*x^5 + 63550*x^4 - 63550*x^3 + 64056*x^2 - 64056*x + 64079)
 
gp: K = bnfinit(x^22 - x^21 + 24*x^20 - 24*x^19 + 254*x^18 - 254*x^17 + 1565*x^16 - 1565*x^15 + 6257*x^14 - 6257*x^13 + 17205*x^12 - 17205*x^11 + 33949*x^10 - 33949*x^9 + 50394*x^8 - 50394*x^7 + 60261*x^6 - 60261*x^5 + 63550*x^4 - 63550*x^3 + 64056*x^2 - 64056*x + 64079, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 24 x^{20} - 24 x^{19} + 254 x^{18} - 254 x^{17} + 1565 x^{16} - 1565 x^{15} + 6257 x^{14} - 6257 x^{13} + 17205 x^{12} - 17205 x^{11} + 33949 x^{10} - 33949 x^{9} + 50394 x^{8} - 50394 x^{7} + 60261 x^{6} - 60261 x^{5} + 63550 x^{4} - 63550 x^{3} + 64056 x^{2} - 64056 x + 64079 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1927323443393334271838358868310546875=-\,5^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.60$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(6,·)$, $\chi_{115}(71,·)$, $\chi_{115}(74,·)$, $\chi_{115}(14,·)$, $\chi_{115}(79,·)$, $\chi_{115}(16,·)$, $\chi_{115}(81,·)$, $\chi_{115}(19,·)$, $\chi_{115}(84,·)$, $\chi_{115}(89,·)$, $\chi_{115}(26,·)$, $\chi_{115}(31,·)$, $\chi_{115}(96,·)$, $\chi_{115}(34,·)$, $\chi_{115}(99,·)$, $\chi_{115}(36,·)$, $\chi_{115}(101,·)$, $\chi_{115}(41,·)$, $\chi_{115}(44,·)$, $\chi_{115}(109,·)$, $\chi_{115}(114,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{28657} a^{12} + \frac{10946}{28657} a^{11} + \frac{12}{28657} a^{10} + \frac{5778}{28657} a^{9} + \frac{54}{28657} a^{8} - \frac{5545}{28657} a^{7} + \frac{112}{28657} a^{6} + \frac{11789}{28657} a^{5} + \frac{105}{28657} a^{4} + \frac{233}{28657} a^{3} + \frac{36}{28657} a^{2} + \frac{5778}{28657} a + \frac{2}{28657}$, $\frac{1}{28657} a^{13} + \frac{13}{28657} a^{11} - \frac{10946}{28657} a^{10} + \frac{65}{28657} a^{9} + \frac{5168}{28657} a^{8} + \frac{156}{28657} a^{7} - \frac{10569}{28657} a^{6} + \frac{182}{28657} a^{5} - \frac{2817}{28657} a^{4} + \frac{91}{28657} a^{3} + \frac{12920}{28657} a^{2} + \frac{13}{28657} a + \frac{6765}{28657}$, $\frac{1}{28657} a^{14} - \frac{9959}{28657} a^{11} - \frac{91}{28657} a^{10} - \frac{12632}{28657} a^{9} - \frac{546}{28657} a^{8} + \frac{4202}{28657} a^{7} - \frac{1274}{28657} a^{6} - \frac{12789}{28657} a^{5} - \frac{1274}{28657} a^{4} + \frac{9891}{28657} a^{3} - \frac{455}{28657} a^{2} - \frac{11035}{28657} a - \frac{26}{28657}$, $\frac{1}{28657} a^{15} - \frac{105}{28657} a^{11} - \frac{7752}{28657} a^{10} - \frac{700}{28657} a^{9} - \frac{2495}{28657} a^{8} - \frac{1890}{28657} a^{7} + \frac{13653}{28657} a^{6} - \frac{2352}{28657} a^{5} - \frac{4723}{28657} a^{4} - \frac{1225}{28657} a^{3} + \frac{3605}{28657} a^{2} - \frac{180}{28657} a - \frac{8739}{28657}$, $\frac{1}{28657} a^{16} - \frac{4702}{28657} a^{11} + \frac{560}{28657} a^{10} + \frac{2398}{28657} a^{9} + \frac{3780}{28657} a^{8} + \frac{4568}{28657} a^{7} + \frac{9408}{28657} a^{6} + \frac{871}{28657} a^{5} + \frac{9800}{28657} a^{4} - \frac{587}{28657} a^{3} + \frac{3600}{28657} a^{2} - \frac{3846}{28657} a + \frac{210}{28657}$, $\frac{1}{28657} a^{17} + \frac{680}{28657} a^{11} + \frac{1508}{28657} a^{10} + \frac{5100}{28657} a^{9} + \frac{563}{28657} a^{8} - \frac{13969}{28657} a^{7} + \frac{11669}{28657} a^{6} - \frac{9617}{28657} a^{5} + \frac{5954}{28657} a^{4} + \frac{10200}{28657} a^{3} - \frac{6516}{28657} a^{2} + \frac{1530}{28657} a + \frac{9404}{28657}$, $\frac{1}{28657} a^{18} + \frac{9048}{28657} a^{11} - \frac{3060}{28657} a^{10} - \frac{2468}{28657} a^{9} + \frac{6625}{28657} a^{8} - \frac{455}{28657} a^{7} + \frac{194}{28657} a^{6} + \frac{13394}{28657} a^{5} - \frac{3886}{28657} a^{4} + \frac{6986}{28657} a^{3} + \frac{5707}{28657} a^{2} + \frac{6373}{28657} a - \frac{1360}{28657}$, $\frac{1}{28657} a^{19} - \frac{3876}{28657} a^{11} + \frac{3584}{28657} a^{10} - \frac{2351}{28657} a^{9} - \frac{1878}{28657} a^{8} - \frac{7053}{28657} a^{7} + \frac{3013}{28657} a^{6} - \frac{9404}{28657} a^{5} + \frac{2627}{28657} a^{4} - \frac{10516}{28657} a^{3} - \frac{4128}{28657} a^{2} - \frac{10336}{28657} a + \frac{10561}{28657}$, $\frac{1}{28657} a^{20} - \frac{10737}{28657} a^{11} - \frac{13153}{28657} a^{10} + \frac{12533}{28657} a^{9} + \frac{1652}{28657} a^{8} + \frac{3343}{28657} a^{7} - \frac{5147}{28657} a^{6} - \frac{11124}{28657} a^{5} - \frac{4734}{28657} a^{4} + \frac{10613}{28657} a^{3} - \frac{14085}{28657} a^{2} - \frac{3685}{28657} a + \frac{7752}{28657}$, $\frac{1}{28657} a^{21} - \frac{8308}{28657} a^{11} - \frac{1908}{28657} a^{10} - \frac{2367}{28657} a^{9} + \frac{10001}{28657} a^{8} + \frac{7434}{28657} a^{7} - \frac{12174}{28657} a^{6} - \frac{4210}{28657} a^{5} - \frac{8282}{28657} a^{4} - \frac{5523}{28657} a^{3} + \frac{10306}{28657} a^{2} + \frac{3733}{28657} a - \frac{7183}{28657}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{662}$, which has order $662$ (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:  $10$
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:  \( 1038656.82438 \) (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

\(\Q(\sqrt{-115}) \), \(\Q(\zeta_{23})^+\)

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 $22$ $22$ R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
5Data not computed
23Data not computed