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Properties

Label 22.2.12677869004...7136.1
Degree $22$
Signature $[2, 10]$
Discriminant $2^{33}\cdot 11^{2}\cdot 121975057867858073$
Root discriminant $21.03$
Ramified primes $2, 11, 121975057867858073$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T57

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 47, -144, 333, -628, 1009, -1416, 1765, -1978, 2008, -1862, 1579, -1228, 876, -572, 341, -184, 89, -38, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 14*x^20 - 38*x^19 + 89*x^18 - 184*x^17 + 341*x^16 - 572*x^15 + 876*x^14 - 1228*x^13 + 1579*x^12 - 1862*x^11 + 2008*x^10 - 1978*x^9 + 1765*x^8 - 1416*x^7 + 1009*x^6 - 628*x^5 + 333*x^4 - 144*x^3 + 47*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^22 - 4*x^21 + 14*x^20 - 38*x^19 + 89*x^18 - 184*x^17 + 341*x^16 - 572*x^15 + 876*x^14 - 1228*x^13 + 1579*x^12 - 1862*x^11 + 2008*x^10 - 1978*x^9 + 1765*x^8 - 1416*x^7 + 1009*x^6 - 628*x^5 + 333*x^4 - 144*x^3 + 47*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} + 14 x^{20} - 38 x^{19} + 89 x^{18} - 184 x^{17} + 341 x^{16} - 572 x^{15} + 876 x^{14} - 1228 x^{13} + 1579 x^{12} - 1862 x^{11} + 2008 x^{10} - 1978 x^{9} + 1765 x^{8} - 1416 x^{7} + 1009 x^{6} - 628 x^{5} + 333 x^{4} - 144 x^{3} + 47 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(126778690041778214971308507136=2^{33}\cdot 11^{2}\cdot 121975057867858073\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.03$
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:  $2, 11, 121975057867858073$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$

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:  $11$
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:  \( 495394.290053 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T57:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3186701844480000
The 1652 conjugacy class representatives for t22n57 are not computed
Character table for t22n57 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $20{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.0.1$x^{12} - x + 7$$1$$12$$0$$C_{12}$$[\ ]^{12}$
121975057867858073Data not computed