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Properties

Label 20.0.40194758889...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 11^{10}$
Root discriminant $47.89$
Ramified primes $2, 3, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 28672, 27648, 89856, 91584, 151776, 110016, 114720, 61092, 54102, 14496, 14160, -1344, 2976, -729, 429, -84, 22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 22*x^18 - 84*x^17 + 429*x^16 - 729*x^15 + 2976*x^14 - 1344*x^13 + 14160*x^12 + 14496*x^11 + 54102*x^10 + 61092*x^9 + 114720*x^8 + 110016*x^7 + 151776*x^6 + 91584*x^5 + 89856*x^4 + 27648*x^3 + 28672*x^2 + 4096)
 
gp: K = bnfinit(x^20 - 3*x^19 + 22*x^18 - 84*x^17 + 429*x^16 - 729*x^15 + 2976*x^14 - 1344*x^13 + 14160*x^12 + 14496*x^11 + 54102*x^10 + 61092*x^9 + 114720*x^8 + 110016*x^7 + 151776*x^6 + 91584*x^5 + 89856*x^4 + 27648*x^3 + 28672*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 22 x^{18} - 84 x^{17} + 429 x^{16} - 729 x^{15} + 2976 x^{14} - 1344 x^{13} + 14160 x^{12} + 14496 x^{11} + 54102 x^{10} + 61092 x^{9} + 114720 x^{8} + 110016 x^{7} + 151776 x^{6} + 91584 x^{5} + 89856 x^{4} + 27648 x^{3} + 28672 x^{2} + 4096 \)

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:  \(4019475888928019955600000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.89$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{3}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{3}{32} a^{10} - \frac{15}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{5}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{3}{64} a^{11} - \frac{15}{64} a^{10} + \frac{15}{32} a^{9} - \frac{3}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{5}{32} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{16} - \frac{1}{128} a^{15} - \frac{3}{128} a^{12} - \frac{15}{128} a^{11} + \frac{15}{64} a^{10} + \frac{5}{16} a^{9} - \frac{5}{16} a^{8} + \frac{1}{4} a^{7} + \frac{27}{64} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{17} - \frac{1}{256} a^{16} - \frac{3}{256} a^{13} - \frac{15}{256} a^{12} + \frac{15}{128} a^{11} + \frac{5}{32} a^{10} + \frac{11}{32} a^{9} + \frac{1}{8} a^{8} - \frac{37}{128} a^{7} - \frac{7}{16} a^{6} - \frac{15}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{534016} a^{18} + \frac{281}{534016} a^{17} - \frac{649}{267008} a^{16} - \frac{101}{33376} a^{15} + \frac{6421}{534016} a^{14} + \frac{15987}{534016} a^{13} + \frac{193}{16688} a^{12} - \frac{13611}{133504} a^{11} + \frac{5}{112} a^{10} - \frac{9683}{33376} a^{9} - \frac{56613}{267008} a^{8} + \frac{31219}{133504} a^{7} - \frac{7781}{66752} a^{6} + \frac{1907}{33376} a^{5} + \frac{177}{1043} a^{4} + \frac{2679}{8344} a^{3} - \frac{1529}{4172} a^{2} + \frac{67}{1043} a - \frac{517}{1043}$, $\frac{1}{202406109381018120348559066901578640384} a^{19} - \frac{21639622708031359545768418951013}{202406109381018120348559066901578640384} a^{18} - \frac{5248162353014240440494162204879529}{3614394810375323577652840480385332864} a^{17} + \frac{11490088825919147003008312372893585}{50601527345254530087139766725394660096} a^{16} - \frac{739412033000452939745910880168454827}{202406109381018120348559066901578640384} a^{15} + \frac{189709588585718976322313703023151357}{202406109381018120348559066901578640384} a^{14} - \frac{297038026666292620485171066012028265}{101203054690509060174279533450789320192} a^{13} - \frac{557752654851081176246930278998983049}{50601527345254530087139766725394660096} a^{12} + \frac{342573230825112239349391406705695091}{25300763672627265043569883362697330048} a^{11} - \frac{3068872609797912998056978631972909145}{12650381836313632521784941681348665024} a^{10} - \frac{9001477165329063778909177769862488613}{101203054690509060174279533450789320192} a^{9} + \frac{998958865820339446613233778005839793}{3614394810375323577652840480385332864} a^{8} - \frac{2447687664399097212288998757117671205}{6325190918156816260892470840674332512} a^{7} + \frac{2333631805816439979220474877838336273}{6325190918156816260892470840674332512} a^{6} + \frac{2329786735432333319081450233885677485}{6325190918156816260892470840674332512} a^{5} + \frac{98107142751761956226514683544212337}{1581297729539204065223117710168583128} a^{4} + \frac{78753443652960876051173163444913663}{1581297729539204065223117710168583128} a^{3} - \frac{45231821422485766610661292219722137}{112949837824228861801651265012041652} a^{2} - \frac{70723670497081715908924279609337449}{197662216192400508152889713771072891} a + \frac{82135357420445737555591225905835024}{197662216192400508152889713771072891}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -1 \) (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:  \( 1172529913.9999409 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}, \sqrt{33})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.21133112220000000.1, 10.2.12679867332000000.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
11Data not computed