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Properties

Label 20.20.7303816416...9296.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18877, -193988, -827226, -1827912, -1985280, -256700, 1829836, 1741890, 1044, -844624, -345938, 138432, 114593, -2978, -16494, -1396, 1252, 138, -52, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 52*x^18 + 138*x^17 + 1252*x^16 - 1396*x^15 - 16494*x^14 - 2978*x^13 + 114593*x^12 + 138432*x^11 - 345938*x^10 - 844624*x^9 + 1044*x^8 + 1741890*x^7 + 1829836*x^6 - 256700*x^5 - 1985280*x^4 - 1827912*x^3 - 827226*x^2 - 193988*x - 18877)
 
gp: K = bnfinit(x^20 - 4*x^19 - 52*x^18 + 138*x^17 + 1252*x^16 - 1396*x^15 - 16494*x^14 - 2978*x^13 + 114593*x^12 + 138432*x^11 - 345938*x^10 - 844624*x^9 + 1044*x^8 + 1741890*x^7 + 1829836*x^6 - 256700*x^5 - 1985280*x^4 - 1827912*x^3 - 827226*x^2 - 193988*x - 18877, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 52 x^{18} + 138 x^{17} + 1252 x^{16} - 1396 x^{15} - 16494 x^{14} - 2978 x^{13} + 114593 x^{12} + 138432 x^{11} - 345938 x^{10} - 844624 x^{9} + 1044 x^{8} + 1741890 x^{7} + 1829836 x^{6} - 256700 x^{5} - 1985280 x^{4} - 1827912 x^{3} - 827226 x^{2} - 193988 x - 18877 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{1}{11} a^{13} + \frac{5}{11} a^{12} - \frac{4}{11} a^{11} + \frac{4}{11} a^{10} - \frac{4}{11} a^{9} - \frac{1}{11} a^{8} - \frac{2}{11} a^{7} + \frac{4}{11} a^{6} - \frac{4}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{14} + \frac{5}{11} a^{13} - \frac{4}{11} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{4}{11} a^{7} - \frac{4}{11} a^{5} + \frac{2}{11} a^{4} - \frac{4}{11} a^{3} + \frac{1}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} - \frac{5}{11} a^{13} - \frac{1}{11} a^{12} - \frac{5}{11} a^{10} + \frac{2}{11} a^{9} + \frac{5}{11} a^{8} + \frac{2}{11} a^{7} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} - \frac{1}{11} a^{3} + \frac{5}{11} a^{2} - \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{14} + \frac{5}{11} a^{13} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} + \frac{4}{11} a^{10} + \frac{3}{11} a^{9} - \frac{4}{11} a^{8} + \frac{2}{11} a^{7} + \frac{4}{11} a^{6} - \frac{3}{11} a^{4} - \frac{5}{11} a^{3} - \frac{3}{11} a^{2} + \frac{5}{11} a - \frac{5}{11}$, $\frac{1}{104586436537534440185213} a^{19} + \frac{3360605681976182169001}{104586436537534440185213} a^{18} + \frac{21856925207066457509}{864350715186235042853} a^{17} + \frac{3319346244771738019141}{104586436537534440185213} a^{16} + \frac{3049333553692520269862}{104586436537534440185213} a^{15} - \frac{12106411502977796912793}{104586436537534440185213} a^{14} + \frac{44764902627980478458998}{104586436537534440185213} a^{13} - \frac{3379911766821242292460}{9507857867048585471383} a^{12} + \frac{26970031621438560924583}{104586436537534440185213} a^{11} + \frac{45179406775506574900883}{104586436537534440185213} a^{10} - \frac{16670422893108578016784}{104586436537534440185213} a^{9} + \frac{14371084575545187333927}{104586436537534440185213} a^{8} + \frac{41862498741147428340607}{104586436537534440185213} a^{7} + \frac{17713285030607090950985}{104586436537534440185213} a^{6} - \frac{2635016673101663343630}{104586436537534440185213} a^{5} - \frac{16026054003375704377163}{104586436537534440185213} a^{4} + \frac{4988197584545178007233}{104586436537534440185213} a^{3} + \frac{29750419738002683474146}{104586436537534440185213} a^{2} + \frac{32872459506913532847732}{104586436537534440185213} a - \frac{1085537829754219330702}{2432242710175219539191}$

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:  $19$
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:  \( 41430036329.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5$ (as 20T17):

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

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.1 x2, 10.10.116117348402176.2

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ 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 ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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

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
2Data not computed
$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$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$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$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$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}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$