Properties

Label 22.2.172...125.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.723\times 10^{27}$
Root discriminant $17.30$
Ramified primes $5, 12917, 459847$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 22T47

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 3*x^20 - 8*x^19 - 13*x^18 - 13*x^17 - 10*x^16 + 79*x^15 + 45*x^14 + 111*x^13 + 59*x^12 - 172*x^11 + 31*x^10 - 94*x^9 - 96*x^8 + 55*x^7 - 26*x^6 + x^5 + 20*x^4 - 4*x^3 + 3*x^2 - 1)
 
gp: K = bnfinit(x^22 + 3*x^20 - 8*x^19 - 13*x^18 - 13*x^17 - 10*x^16 + 79*x^15 + 45*x^14 + 111*x^13 + 59*x^12 - 172*x^11 + 31*x^10 - 94*x^9 - 96*x^8 + 55*x^7 - 26*x^6 + x^5 + 20*x^4 - 4*x^3 + 3*x^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 3, -4, 20, 1, -26, 55, -96, -94, 31, -172, 59, 111, 45, 79, -10, -13, -13, -8, 3, 0, 1]);
 

\( x^{22} + 3 x^{20} - 8 x^{19} - 13 x^{18} - 13 x^{17} - 10 x^{16} + 79 x^{15} + 45 x^{14} + 111 x^{13} + 59 x^{12} - 172 x^{11} + 31 x^{10} - 94 x^{9} - 96 x^{8} + 55 x^{7} - 26 x^{6} + x^{5} + 20 x^{4} - 4 x^{3} + 3 x^{2} - 1 \)

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1722741365651855595751953125\)\(\medspace = 5^{11}\cdot 12917^{2}\cdot 459847^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.30$
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:  $5, 12917, 459847$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2401618190265284674811} a^{21} - \frac{14244213809825361172}{2401618190265284674811} a^{20} - \frac{1002104084110697076513}{2401618190265284674811} a^{19} - \frac{367216667644907231650}{2401618190265284674811} a^{18} + \frac{229059533055962844563}{2401618190265284674811} a^{17} + \frac{399176911272701937276}{2401618190265284674811} a^{16} + \frac{758472351089832778349}{2401618190265284674811} a^{15} - \frac{177642923876279655337}{2401618190265284674811} a^{14} + \frac{103312316322764112376}{2401618190265284674811} a^{13} + \frac{1110083896221763476653}{2401618190265284674811} a^{12} + \frac{699186610257392640747}{2401618190265284674811} a^{11} - \frac{812554669888071682978}{2401618190265284674811} a^{10} - \frac{15297580225974447004}{2401618190265284674811} a^{9} + \frac{994462095366045419597}{2401618190265284674811} a^{8} - \frac{1028122001191647940185}{2401618190265284674811} a^{7} - \frac{913831922552747611054}{2401618190265284674811} a^{6} - \frac{1091232064227718910422}{2401618190265284674811} a^{5} + \frac{853402051447813984393}{2401618190265284674811} a^{4} + \frac{501990338122522928382}{2401618190265284674811} a^{3} - \frac{473625369548173765519}{2401618190265284674811} a^{2} + \frac{848474124628934939444}{2401618190265284674811} a + \frac{694785389619438405661}{2401618190265284674811}$

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:  $11$
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:  \( 42078.318318 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{10}\cdot 42078.318318 \cdot 1}{2\sqrt{1722741365651855595751953125}}\approx 0.19443614995$ (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

\(\Q(\sqrt{5}) \), 11.1.5939843699.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ $22$ R $22$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
12917Data not computed
459847Data not computed