Properties

Label 22.0.110...347.1
Degree $22$
Signature $[0, 11]$
Discriminant $-1.104\times 10^{26}$
Root discriminant $15.27$
Ramified primes $3, 971, 25709231$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 22T47

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1)
 
gp: K = bnfinit(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 4, 1, 6, 5, 13, 4, 18, 2, 12, -3, 7, -6, 1, -4, 4, 0, 0, -4, 0, 0, 1]);
 

\( x^{22} - 4 x^{19} + 4 x^{16} - 4 x^{15} + x^{14} - 6 x^{13} + 7 x^{12} - 3 x^{11} + 12 x^{10} + 2 x^{9} + 18 x^{8} + 4 x^{7} + 13 x^{6} + 5 x^{5} + 6 x^{4} + x^{3} + 4 x^{2} - x + 1 \)

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:  \(-110395262036162513388217347\)\(\medspace = -\,3^{11}\cdot 971^{2}\cdot 25709231^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.27$
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:  $3, 971, 25709231$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{3835646} a^{21} + \frac{160649}{3835646} a^{20} - \frac{21455}{1917823} a^{19} - \frac{396366}{1917823} a^{18} + \frac{1433247}{3835646} a^{17} + \frac{810696}{1917823} a^{16} - \frac{1798625}{3835646} a^{15} - \frac{423157}{3835646} a^{14} - \frac{297417}{1917823} a^{13} - \frac{120651}{3835646} a^{12} + \frac{974569}{3835646} a^{11} - \frac{1780973}{3835646} a^{10} - \frac{553605}{1917823} a^{9} + \frac{150493}{3835646} a^{8} + \frac{473237}{3835646} a^{7} - \frac{1288549}{3835646} a^{6} - \frac{47137}{3835646} a^{5} - \frac{473352}{1917823} a^{4} + \frac{74328}{1917823} a^{3} + \frac{705749}{3835646} a^{2} + \frac{10991}{3835646} a - \frac{621825}{3835646}$

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:  \( -\frac{2446799}{3835646} a^{21} - \frac{364147}{1917823} a^{20} + \frac{924955}{3835646} a^{19} + \frac{9956951}{3835646} a^{18} + \frac{2441111}{3835646} a^{17} - \frac{1997558}{1917823} a^{16} - \frac{5101796}{1917823} a^{15} + \frac{8857079}{3835646} a^{14} + \frac{5279489}{3835646} a^{13} + \frac{9758697}{3835646} a^{12} - \frac{14706767}{3835646} a^{11} - \frac{2136465}{3835646} a^{10} - \frac{9731443}{1917823} a^{9} - \frac{6847511}{1917823} a^{8} - \frac{37853405}{3835646} a^{7} - \frac{10577418}{1917823} a^{6} - \frac{10185582}{1917823} a^{5} - \frac{8712443}{1917823} a^{4} - \frac{6192274}{1917823} a^{3} - \frac{6611313}{3835646} a^{2} - \frac{3403586}{1917823} a + \frac{421412}{1917823} \) (order $6$)
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:  \( 18267.9467933 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 18267.9467933 \cdot 1}{6\sqrt{110395262036162513388217347}}\approx 0.174599028244$ (assuming GRH)

Galois group

22T47:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.3.24963663301.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R $22$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $22$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. 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
3Data not computed
971Data not computed
25709231Data not computed