Properties

Label 20.4.359...241.1
Degree $20$
Signature $[4, 8]$
Discriminant $3.598\times 10^{27}$
Root discriminant $23.87$
Ramified primes $11, 23$
Class number $2$
Class group $[2]$
Galois group $C_2^4:C_5$ (as 20T23)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -20, 86, -205, 386, -572, 696, -703, 548, -413, 205, -162, 71, -106, 60, -77, 27, -21, 3, -4, 1]);
 

\( x^{20} - 4 x^{19} + 3 x^{18} - 21 x^{17} + 27 x^{16} - 77 x^{15} + 60 x^{14} - 106 x^{13} + 71 x^{12} - 162 x^{11} + 205 x^{10} - 413 x^{9} + 548 x^{8} - 703 x^{7} + 696 x^{6} - 572 x^{5} + 386 x^{4} - 205 x^{3} + 86 x^{2} - 20 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3598368619012125638152362241\)\(\medspace = 11^{16}\cdot 23^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.87$
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:  $11, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{17} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{66} a^{18} - \frac{1}{66} a^{17} - \frac{5}{66} a^{16} + \frac{1}{33} a^{15} + \frac{7}{33} a^{14} - \frac{2}{11} a^{13} + \frac{10}{33} a^{12} - \frac{5}{11} a^{11} + \frac{13}{66} a^{10} + \frac{5}{66} a^{9} + \frac{23}{66} a^{8} + \frac{8}{33} a^{7} + \frac{19}{66} a^{6} - \frac{1}{6} a^{5} + \frac{3}{11} a^{4} + \frac{5}{33} a^{3} - \frac{2}{33} a^{2} - \frac{1}{22} a + \frac{10}{33}$, $\frac{1}{18914411302141938} a^{19} - \frac{20267615738053}{3152401883690323} a^{18} - \frac{565599225167281}{9457205651070969} a^{17} - \frac{39474093925372}{3152401883690323} a^{16} + \frac{319876137990509}{6304803767380646} a^{15} - \frac{1518414626185591}{3152401883690323} a^{14} - \frac{3101651311932845}{9457205651070969} a^{13} - \frac{705392634521349}{3152401883690323} a^{12} + \frac{4175792663419093}{18914411302141938} a^{11} - \frac{4120896052077995}{9457205651070969} a^{10} - \frac{2757808332678745}{9457205651070969} a^{9} - \frac{948462612611955}{3152401883690323} a^{8} - \frac{927809049955432}{9457205651070969} a^{7} + \frac{79182244307674}{286581989426393} a^{6} + \frac{4826086194566831}{18914411302141938} a^{5} - \frac{4287814152720055}{18914411302141938} a^{4} - \frac{4875444187727579}{18914411302141938} a^{3} - \frac{2673366357731461}{9457205651070969} a^{2} + \frac{6632957682809995}{18914411302141938} a + \frac{65333108471271}{573163978852786}$

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

Class group and class number

$C_{2}$, which has order $2$

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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 436538.984162 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{8}\cdot 436538.984162 \cdot 2}{2\sqrt{3598368619012125638152362241}}\approx 0.282832386255$

Galois group

$C_2^4:C_5$ (as 20T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.113395848049.1, 10.6.113395848049.1, 10.6.59986403617921.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$