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Properties

Label 20.8.27866822111...3321.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{22}\cdot 11^{8}\cdot 23^{10}$
Root discriminant $41.90$
Ramified primes $3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 0, 847, 0, 770, 0, -2507, 0, -2874, 0, 439, 0, 938, 0, -21, 0, -79, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 79*x^16 - 21*x^14 + 938*x^12 + 439*x^10 - 2874*x^8 - 2507*x^6 + 770*x^4 + 847*x^2 + 121)
 
gp: K = bnfinit(x^20 + 2*x^18 - 79*x^16 - 21*x^14 + 938*x^12 + 439*x^10 - 2874*x^8 - 2507*x^6 + 770*x^4 + 847*x^2 + 121, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 79 x^{16} - 21 x^{14} + 938 x^{12} + 439 x^{10} - 2874 x^{8} - 2507 x^{6} + 770 x^{4} + 847 x^{2} + 121 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(278668221112378935712597132533321=3^{22}\cdot 11^{8}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.90$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{66} a^{15} - \frac{1}{66} a^{13} - \frac{5}{33} a^{11} - \frac{13}{66} a^{9} - \frac{1}{2} a^{8} + \frac{3}{22} a^{7} - \frac{1}{2} a^{6} - \frac{14}{33} a^{5} - \frac{29}{66} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{66} a^{16} - \frac{1}{66} a^{14} - \frac{5}{33} a^{12} - \frac{13}{66} a^{10} - \frac{1}{2} a^{9} + \frac{3}{22} a^{8} - \frac{1}{2} a^{7} - \frac{14}{33} a^{6} - \frac{29}{66} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{66} a^{17} - \frac{1}{6} a^{13} + \frac{5}{33} a^{11} + \frac{29}{66} a^{9} + \frac{7}{33} a^{7} - \frac{4}{11} a^{5} + \frac{13}{33} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{2840874167538} a^{18} + \frac{4717830811}{2840874167538} a^{16} + \frac{113559177709}{2840874167538} a^{14} - \frac{31312670476}{129130643979} a^{12} - \frac{97299251381}{946958055846} a^{10} + \frac{288188767400}{1420437083769} a^{8} - \frac{1}{2} a^{7} + \frac{1297186557395}{2840874167538} a^{6} - \frac{1353808368949}{2840874167538} a^{4} - \frac{12320916071}{43043547993} a^{2} - \frac{40252063259}{129130643979}$, $\frac{1}{2840874167538} a^{19} + \frac{4717830811}{2840874167538} a^{17} - \frac{7785733135}{1420437083769} a^{15} - \frac{559748106493}{2840874167538} a^{13} - \frac{70171399687}{473479027923} a^{11} + \frac{417319411379}{1420437083769} a^{9} - \frac{1}{2} a^{8} + \frac{67505380792}{1420437083769} a^{7} - \frac{579024505075}{2840874167538} a^{5} - \frac{1}{2} a^{4} + \frac{30244682389}{946958055846} a^{3} + \frac{48626517461}{258261287958} a$

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:  $13$
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:  \( 761281923.777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{69}) \), 5.5.5184729.1, 10.4.80644244410323.1, 10.4.5564452864312287.1, 10.10.16693358592936861.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.18.85$x^{12} + 36 x^{11} + 111 x^{10} + 90 x^{9} + 36 x^{8} + 90 x^{7} + 30 x^{6} + 108 x^{5} - 36 x^{4} + 54 x^{3} - 81 x^{2} + 54 x - 18$$6$$2$$18$$D_6$$[2]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.4.2.1$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}$
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}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$