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Properties

Label 20.0.13782929168...3904.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{12}\cdot 3^{10}\cdot 11^{8}\cdot 113^{8}$
Root discriminant $45.39$
Ramified primes $2, 3, 11, 113$
Class number $108$ (GRH)
Class group $[6, 18]$ (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 46, 0, 2019, 0, 4340, 0, 6589, 0, 4630, 0, 2317, 0, 660, 0, 135, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 14*x^18 + 135*x^16 + 660*x^14 + 2317*x^12 + 4630*x^10 + 6589*x^8 + 4340*x^6 + 2019*x^4 + 46*x^2 + 1)
 
gp: K = bnfinit(x^20 + 14*x^18 + 135*x^16 + 660*x^14 + 2317*x^12 + 4630*x^10 + 6589*x^8 + 4340*x^6 + 2019*x^4 + 46*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 14 x^{18} + 135 x^{16} + 660 x^{14} + 2317 x^{12} + 4630 x^{10} + 6589 x^{8} + 4340 x^{6} + 2019 x^{4} + 46 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1378292916885385650670001824763904=2^{12}\cdot 3^{10}\cdot 11^{8}\cdot 113^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.39$
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:  $2, 3, 11, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{28} a^{16} + \frac{1}{14} a^{14} + \frac{1}{14} a^{12} - \frac{1}{4} a^{11} + \frac{5}{28} a^{10} - \frac{1}{4} a^{9} - \frac{5}{28} a^{8} - \frac{1}{4} a^{7} + \frac{1}{28} a^{6} + \frac{1}{4} a^{5} + \frac{5}{14} a^{4} + \frac{1}{4} a^{3} - \frac{5}{28} a^{2} + \frac{1}{4} a + \frac{9}{28}$, $\frac{1}{56} a^{17} + \frac{1}{28} a^{15} - \frac{1}{8} a^{14} - \frac{5}{56} a^{13} - \frac{9}{56} a^{11} + \frac{1}{8} a^{10} + \frac{9}{56} a^{9} - \frac{1}{8} a^{8} - \frac{13}{56} a^{7} - \frac{3}{8} a^{6} + \frac{3}{7} a^{5} + \frac{3}{8} a^{4} - \frac{19}{56} a^{3} + \frac{1}{4} a^{2} - \frac{13}{28} a + \frac{3}{8}$, $\frac{1}{2220605550632} a^{18} + \frac{2294666107}{277575693829} a^{16} - \frac{1}{8} a^{15} + \frac{257402713613}{2220605550632} a^{14} + \frac{269515116401}{2220605550632} a^{12} + \frac{1}{8} a^{11} - \frac{86886629731}{2220605550632} a^{10} - \frac{1}{8} a^{9} + \frac{275519263959}{2220605550632} a^{8} + \frac{1}{8} a^{7} - \frac{36745445449}{79307341094} a^{6} - \frac{1}{8} a^{5} + \frac{48141119625}{317229364376} a^{4} - \frac{1}{4} a^{3} + \frac{253974903629}{555151387658} a^{2} + \frac{3}{8} a - \frac{294083528105}{1110302775316}$, $\frac{1}{2220605550632} a^{19} + \frac{2294666107}{277575693829} a^{17} - \frac{1}{56} a^{16} + \frac{257402713613}{2220605550632} a^{15} - \frac{1}{28} a^{14} + \frac{269515116401}{2220605550632} a^{13} + \frac{5}{56} a^{12} - \frac{86886629731}{2220605550632} a^{11} + \frac{9}{56} a^{10} + \frac{275519263959}{2220605550632} a^{9} - \frac{9}{56} a^{8} + \frac{1454112549}{39653670547} a^{7} - \frac{15}{56} a^{6} - \frac{110473562563}{317229364376} a^{5} - \frac{3}{7} a^{4} - \frac{11800395100}{277575693829} a^{3} + \frac{19}{56} a^{2} - \frac{294083528105}{1110302775316} a - \frac{1}{28}$

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

Class group and class number

$C_{6}\times C_{18}$, which has order $108$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{923400373}{39653670547} a^{18} + \frac{51587174001}{158614682188} a^{16} + \frac{124245233665}{39653670547} a^{14} + \frac{2422067090135}{158614682188} a^{12} + \frac{8485543353779}{158614682188} a^{10} + \frac{16855956378065}{158614682188} a^{8} + \frac{23904161809423}{158614682188} a^{6} + \frac{3865253324013}{39653670547} a^{4} + \frac{7302810785543}{158614682188} a^{2} + \frac{83191590111}{79307341094} \) (order $6$)
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:  \( 6459109.84372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.6180196.1, 10.0.37125367565660352.1, 10.0.9281341891415088.1, 10.10.152779290393664.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$113$113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.6.4.1$x^{6} + 3277 x^{3} + 12769000$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
113.6.4.1$x^{6} + 3277 x^{3} + 12769000$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$