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Properties

Label 20.8.10779662339...2544.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{18}\cdot 43^{2}$
Root discriminant $25.21$
Ramified primes $2, 11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -110, 277, 946, -227, -2288, -256, 2926, 731, -1804, -89, 924, 20, -132, 51, -22, -19, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 19*x^16 - 22*x^15 + 51*x^14 - 132*x^13 + 20*x^12 + 924*x^11 - 89*x^10 - 1804*x^9 + 731*x^8 + 2926*x^7 - 256*x^6 - 2288*x^5 - 227*x^4 + 946*x^3 + 277*x^2 - 110*x - 43)
 
gp: K = bnfinit(x^20 - 5*x^18 - 19*x^16 - 22*x^15 + 51*x^14 - 132*x^13 + 20*x^12 + 924*x^11 - 89*x^10 - 1804*x^9 + 731*x^8 + 2926*x^7 - 256*x^6 - 2288*x^5 - 227*x^4 + 946*x^3 + 277*x^2 - 110*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 19 x^{16} - 22 x^{15} + 51 x^{14} - 132 x^{13} + 20 x^{12} + 924 x^{11} - 89 x^{10} - 1804 x^{9} + 731 x^{8} + 2926 x^{7} - 256 x^{6} - 2288 x^{5} - 227 x^{4} + 946 x^{3} + 277 x^{2} - 110 x - 43 \)

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:  \(10779662339431083287111532544=2^{20}\cdot 11^{18}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.21$
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:  $2, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{89} a^{17} + \frac{8}{89} a^{16} + \frac{33}{89} a^{15} - \frac{5}{89} a^{14} + \frac{19}{89} a^{13} + \frac{27}{89} a^{12} - \frac{11}{89} a^{11} - \frac{34}{89} a^{10} - \frac{11}{89} a^{9} - \frac{12}{89} a^{8} + \frac{39}{89} a^{7} + \frac{25}{89} a^{6} + \frac{26}{89} a^{5} + \frac{16}{89} a^{4} - \frac{15}{89} a^{3} + \frac{40}{89} a^{2} + \frac{41}{89} a - \frac{8}{89}$, $\frac{1}{3827} a^{18} + \frac{1}{3827} a^{17} + \frac{1401}{3827} a^{16} - \frac{1838}{3827} a^{15} + \frac{588}{3827} a^{14} + \frac{1229}{3827} a^{13} + \frac{1046}{3827} a^{12} + \frac{1467}{3827} a^{11} - \frac{1375}{3827} a^{10} - \frac{736}{3827} a^{9} + \frac{924}{3827} a^{8} - \frac{337}{3827} a^{7} - \frac{416}{3827} a^{6} - \frac{1501}{3827} a^{5} + \frac{1831}{3827} a^{4} + \frac{1035}{3827} a^{3} + \frac{28}{3827} a^{2} + \frac{1663}{3827} a + \frac{22}{89}$, $\frac{1}{75611712444368176561} a^{19} + \frac{8846996360004229}{75611712444368176561} a^{18} - \frac{295126463007135742}{75611712444368176561} a^{17} - \frac{19676882708672206726}{75611712444368176561} a^{16} + \frac{35885986802682935231}{75611712444368176561} a^{15} - \frac{2699692291651534753}{75611712444368176561} a^{14} - \frac{8617554806107782132}{75611712444368176561} a^{13} - \frac{36351496475057954949}{75611712444368176561} a^{12} + \frac{5258215220839431707}{75611712444368176561} a^{11} + \frac{6530331104858361032}{75611712444368176561} a^{10} + \frac{1910510432706900306}{75611712444368176561} a^{9} - \frac{31539579982855565964}{75611712444368176561} a^{8} + \frac{8714352614914811485}{75611712444368176561} a^{7} + \frac{12653129395457433985}{75611712444368176561} a^{6} - \frac{1377286614926734722}{75611712444368176561} a^{5} - \frac{31602012164209242590}{75611712444368176561} a^{4} - \frac{34987177611677818951}{75611712444368176561} a^{3} - \frac{34418288642265288555}{75611712444368176561} a^{2} - \frac{15345723744933310959}{75611712444368176561} a - \frac{747706120140470859}{1758411917310887827}$

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:  \( 2536535.5255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
11Data not computed
43Data not computed