↑

Properties

Label 22.22.1478612446...8125.1
Degree $22$
Signature $[22, 0]$
Discriminant $5^{11}\cdot 55029067682009^{2}$
Root discriminant $39.69$
Ramified primes $5, 55029067682009$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T47

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, 31, -126, -402, 1048, 2728, -3533, -9307, 5347, 16648, -3399, -16217, 378, 8835, 491, -2682, -199, 435, 25, -34, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 34*x^20 + 25*x^19 + 435*x^18 - 199*x^17 - 2682*x^16 + 491*x^15 + 8835*x^14 + 378*x^13 - 16217*x^12 - 3399*x^11 + 16648*x^10 + 5347*x^9 - 9307*x^8 - 3533*x^7 + 2728*x^6 + 1048*x^5 - 402*x^4 - 126*x^3 + 31*x^2 + 5*x - 1)
 
gp: K = bnfinit(x^22 - x^21 - 34*x^20 + 25*x^19 + 435*x^18 - 199*x^17 - 2682*x^16 + 491*x^15 + 8835*x^14 + 378*x^13 - 16217*x^12 - 3399*x^11 + 16648*x^10 + 5347*x^9 - 9307*x^8 - 3533*x^7 + 2728*x^6 + 1048*x^5 - 402*x^4 - 126*x^3 + 31*x^2 + 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 34 x^{20} + 25 x^{19} + 435 x^{18} - 199 x^{17} - 2682 x^{16} + 491 x^{15} + 8835 x^{14} + 378 x^{13} - 16217 x^{12} - 3399 x^{11} + 16648 x^{10} + 5347 x^{9} - 9307 x^{8} - 3533 x^{7} + 2728 x^{6} + 1048 x^{5} - 402 x^{4} - 126 x^{3} + 31 x^{2} + 5 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(147861244626519891422962699267578125=5^{11}\cdot 55029067682009^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.69$
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:  $5, 55029067682009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{8050295720193493799} a^{21} + \frac{1682386493903915660}{8050295720193493799} a^{20} + \frac{2028828168194941696}{8050295720193493799} a^{19} + \frac{3531625238481394667}{8050295720193493799} a^{18} - \frac{361962629852261712}{8050295720193493799} a^{17} - \frac{2539069664680220411}{8050295720193493799} a^{16} + \frac{1026516189107871061}{8050295720193493799} a^{15} + \frac{2709000362149847079}{8050295720193493799} a^{14} - \frac{596459041536382312}{8050295720193493799} a^{13} + \frac{1601554111015961375}{8050295720193493799} a^{12} + \frac{3245431849379078147}{8050295720193493799} a^{11} - \frac{629972969538100796}{8050295720193493799} a^{10} + \frac{3009883360880741732}{8050295720193493799} a^{9} - \frac{2689202427751411545}{8050295720193493799} a^{8} - \frac{2456757709921666718}{8050295720193493799} a^{7} + \frac{2346925194982333096}{8050295720193493799} a^{6} + \frac{1787272726581748534}{8050295720193493799} a^{5} + \frac{3929164802240844235}{8050295720193493799} a^{4} + \frac{3875140283713926637}{8050295720193493799} a^{3} + \frac{996111929198540209}{8050295720193493799} a^{2} - \frac{151017421680822803}{8050295720193493799} a - \frac{2614092534560446269}{8050295720193493799}$

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:  $21$
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:  \( 26137547532.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{5}) \), 11.11.55029067682009.1

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 $22$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $22$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
55029067682009Data not computed