Properties

Label 22.0.578...471.1
Degree $22$
Signature $[0, 11]$
Discriminant $-5.790\times 10^{26}$
Root discriminant $16.46$
Ramified prime $271$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{11}$ (as 22T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9)
 
gp: K = bnfinit(x^22 - 11*x^21 + 67*x^20 - 285*x^19 + 929*x^18 - 2433*x^17 + 5271*x^16 - 9630*x^15 + 15091*x^14 - 20633*x^13 + 25057*x^12 - 27453*x^11 + 27302*x^10 - 24426*x^9 + 19254*x^8 - 13050*x^7 + 7570*x^6 - 3890*x^5 + 1921*x^4 - 927*x^3 + 362*x^2 - 87*x + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -87, 362, -927, 1921, -3890, 7570, -13050, 19254, -24426, 27302, -27453, 25057, -20633, 15091, -9630, 5271, -2433, 929, -285, 67, -11, 1]);
 

\( x^{22} - 11 x^{21} + 67 x^{20} - 285 x^{19} + 929 x^{18} - 2433 x^{17} + 5271 x^{16} - 9630 x^{15} + 15091 x^{14} - 20633 x^{13} + 25057 x^{12} - 27453 x^{11} + 27302 x^{10} - 24426 x^{9} + 19254 x^{8} - 13050 x^{7} + 7570 x^{6} - 3890 x^{5} + 1921 x^{4} - 927 x^{3} + 362 x^{2} - 87 x + 9 \)

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-578978183833808423828407471\)\(\medspace = -\,271^{11}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.46$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $271$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $22$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8}$, $\frac{1}{3} a^{17} - \frac{1}{3} a$, $\frac{1}{27} a^{18} - \frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{2}{27} a^{10} + \frac{1}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{10}{27} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{81} a^{19} + \frac{1}{81} a^{18} + \frac{2}{27} a^{17} - \frac{1}{9} a^{16} - \frac{2}{27} a^{15} + \frac{1}{27} a^{14} + \frac{4}{27} a^{13} + \frac{7}{81} a^{11} + \frac{1}{81} a^{10} - \frac{4}{27} a^{9} - \frac{7}{27} a^{7} + \frac{8}{27} a^{6} + \frac{5}{27} a^{5} + \frac{1}{3} a^{4} - \frac{26}{81} a^{3} + \frac{34}{81} a^{2} - \frac{4}{27} a - \frac{4}{9}$, $\frac{1}{148473} a^{20} - \frac{10}{148473} a^{19} + \frac{2560}{148473} a^{18} - \frac{7585}{49491} a^{17} + \frac{4171}{49491} a^{16} + \frac{6152}{49491} a^{15} - \frac{5677}{49491} a^{14} - \frac{587}{3807} a^{13} - \frac{24644}{148473} a^{12} - \frac{2884}{148473} a^{11} + \frac{4027}{148473} a^{10} - \frac{5464}{49491} a^{9} - \frac{9223}{49491} a^{8} + \frac{9751}{49491} a^{7} + \frac{21301}{49491} a^{6} - \frac{19351}{49491} a^{5} - \frac{515}{11421} a^{4} + \frac{4829}{148473} a^{3} - \frac{22877}{148473} a^{2} - \frac{20704}{49491} a - \frac{5644}{16497}$, $\frac{1}{148473} a^{21} + \frac{209}{49491} a^{19} + \frac{1012}{148473} a^{18} + \frac{2380}{16497} a^{17} + \frac{430}{5499} a^{16} - \frac{6479}{49491} a^{15} - \frac{82}{16497} a^{14} + \frac{21376}{148473} a^{13} - \frac{623}{49491} a^{12} + \frac{3949}{49491} a^{11} + \frac{22045}{148473} a^{10} - \frac{7040}{49491} a^{9} + \frac{5501}{16497} a^{8} - \frac{334}{49491} a^{7} + \frac{4675}{16497} a^{6} - \frac{20828}{148473} a^{5} - \frac{20707}{49491} a^{4} + \frac{2620}{16497} a^{3} - \frac{56258}{148473} a^{2} - \frac{2179}{49491} a + \frac{383}{16497}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 33482.7947011 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 33482.7947011 \cdot 1}{2\sqrt{578978183833808423828407471}}\approx 0.419217261313$ (assuming GRH)

Galois group

$D_{11}$ (as 22T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

Intermediate fields

\(\Q(\sqrt{-271}) \), 11.1.1461660310351.1 x11

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

Sibling fields

Degree 11 sibling: 11.1.1461660310351.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
271Data not computed