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Properties

Label 20.20.2588665440...5312.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 17^{15}\cdot 53^{14}$
Root discriminant $234.78$
Ramified primes $2, 17, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4:F_5$ (as 20T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25224604, -321816380, 577682706, 491417950, -1481772709, 182743098, 1107126565, -469380944, -295344111, 194714572, 22954298, -31913924, 1504150, 2366808, -279760, -81202, 12703, 1138, -211, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 211*x^18 + 1138*x^17 + 12703*x^16 - 81202*x^15 - 279760*x^14 + 2366808*x^13 + 1504150*x^12 - 31913924*x^11 + 22954298*x^10 + 194714572*x^9 - 295344111*x^8 - 469380944*x^7 + 1107126565*x^6 + 182743098*x^5 - 1481772709*x^4 + 491417950*x^3 + 577682706*x^2 - 321816380*x + 25224604)
 
gp: K = bnfinit(x^20 - 4*x^19 - 211*x^18 + 1138*x^17 + 12703*x^16 - 81202*x^15 - 279760*x^14 + 2366808*x^13 + 1504150*x^12 - 31913924*x^11 + 22954298*x^10 + 194714572*x^9 - 295344111*x^8 - 469380944*x^7 + 1107126565*x^6 + 182743098*x^5 - 1481772709*x^4 + 491417950*x^3 + 577682706*x^2 - 321816380*x + 25224604, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 211 x^{18} + 1138 x^{17} + 12703 x^{16} - 81202 x^{15} - 279760 x^{14} + 2366808 x^{13} + 1504150 x^{12} - 31913924 x^{11} + 22954298 x^{10} + 194714572 x^{9} - 295344111 x^{8} - 469380944 x^{7} + 1107126565 x^{6} + 182743098 x^{5} - 1481772709 x^{4} + 491417950 x^{3} + 577682706 x^{2} - 321816380 x + 25224604 \)

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:  \(258866544008047715925279948537661096282306445312=2^{16}\cdot 17^{15}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $234.78$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{5}$, $\frac{1}{152} a^{18} - \frac{5}{152} a^{17} - \frac{5}{152} a^{16} - \frac{7}{76} a^{15} + \frac{1}{19} a^{14} + \frac{4}{19} a^{13} - \frac{3}{19} a^{12} + \frac{3}{38} a^{11} - \frac{7}{76} a^{10} - \frac{5}{76} a^{9} + \frac{13}{76} a^{8} - \frac{9}{19} a^{7} + \frac{75}{152} a^{6} + \frac{41}{152} a^{5} + \frac{47}{152} a^{4} + \frac{5}{38} a^{3} - \frac{11}{76} a^{2} + \frac{8}{19} a - \frac{15}{38}$, $\frac{1}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{19} + \frac{28241384478332113334724842276798637547051980057920661835894712620986117841}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{18} - \frac{218372159626874676303006018341200788038210763049929958822956979799293491227}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{17} - \frac{250208025900432150687499393623566425790339571098815262773706022353326873163}{2222505851020359481182833121928214658163331584005300571445389407698060223468} a^{16} - \frac{271010242616542636237520591720656831682839715971832894754124000923911254805}{3333758776530539221774249682892321987244997376007950857168084111547090335202} a^{15} + \frac{207583542497911573231136068715820341745212500212176191376065050917349742874}{1666879388265269610887124841446160993622498688003975428584042055773545167601} a^{14} - \frac{469791552480488659386056790709804911975965065951306412201988745241360405129}{3333758776530539221774249682892321987244997376007950857168084111547090335202} a^{13} + \frac{594264359152046602805881178565736818968258531699358685435793486380611326075}{3333758776530539221774249682892321987244997376007950857168084111547090335202} a^{12} + \frac{1314783368616337427610441819697451979082543032590691845669717743466484742513}{6667517553061078443548499365784643974489994752015901714336168223094180670404} a^{11} - \frac{542764655929711947635941237887118975650129887393162509284533218263468173657}{6667517553061078443548499365784643974489994752015901714336168223094180670404} a^{10} + \frac{555508107964399539993448446022022065574860001886304971191018823264922057775}{2222505851020359481182833121928214658163331584005300571445389407698060223468} a^{9} + \frac{493628530313923538000561251339993431075159235910769602098603586104798702417}{3333758776530539221774249682892321987244997376007950857168084111547090335202} a^{8} - \frac{4898648627054533611064853849853636208099844407482785464846822570809751385857}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{7} + \frac{2265581735077709161996423784325657605479704014568250419572764002201397661003}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{6} + \frac{4210904668891709962555177636813501047465818770698501319399513421458353729149}{13335035106122156887096998731569287948979989504031803428672336446188361340808} a^{5} - \frac{270919352124786161382375406506711281662180640377935005467259644157780321005}{1666879388265269610887124841446160993622498688003975428584042055773545167601} a^{4} - \frac{701014017159739200012516361111045517470433845658873172507471658575003437749}{6667517553061078443548499365784643974489994752015901714336168223094180670404} a^{3} - \frac{75529887823139405395219329251786062660920827187165819377458456449686241055}{370417641836726580197138853654702443027221930667550095240898234616343370578} a^{2} - \frac{361539671004566708603165977563330800704631239936529668597618189435179979259}{1111252925510179740591416560964107329081665792002650285722694703849030111734} a + \frac{20477132595211579069066462150625970403651227891265494637220266111368976268}{87730494119224716362480254812955841769605194105472390978107476619660271979}$

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:  $19$
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:  \( 1146610929550000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:F_5$ (as 20T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_4:F_5$
Character table for $C_4:F_5$

Intermediate fields

\(\Q(\sqrt{901}) \), 4.4.13800617.1, 5.5.2382032.1, 10.10.426987989728139087104.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed