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Properties

Label 20.8.36398622315...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}$
Root discriminant $53.46$
Ramified primes $5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-539, 7721, -30355, 48816, -36533, 28777, -43465, 34606, -12004, 7494, -5928, 2799, -2553, 1576, -703, 340, -50, 9, -5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 5*x^18 + 9*x^17 - 50*x^16 + 340*x^15 - 703*x^14 + 1576*x^13 - 2553*x^12 + 2799*x^11 - 5928*x^10 + 7494*x^9 - 12004*x^8 + 34606*x^7 - 43465*x^6 + 28777*x^5 - 36533*x^4 + 48816*x^3 - 30355*x^2 + 7721*x - 539)
 
gp: K = bnfinit(x^20 - 3*x^19 - 5*x^18 + 9*x^17 - 50*x^16 + 340*x^15 - 703*x^14 + 1576*x^13 - 2553*x^12 + 2799*x^11 - 5928*x^10 + 7494*x^9 - 12004*x^8 + 34606*x^7 - 43465*x^6 + 28777*x^5 - 36533*x^4 + 48816*x^3 - 30355*x^2 + 7721*x - 539, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 5 x^{18} + 9 x^{17} - 50 x^{16} + 340 x^{15} - 703 x^{14} + 1576 x^{13} - 2553 x^{12} + 2799 x^{11} - 5928 x^{10} + 7494 x^{9} - 12004 x^{8} + 34606 x^{7} - 43465 x^{6} + 28777 x^{5} - 36533 x^{4} + 48816 x^{3} - 30355 x^{2} + 7721 x - 539 \)

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:  \(36398622315033148150802886962890625=5^{15}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.46$
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:  $5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{15} - \frac{1}{7} a^{14} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{147} a^{18} + \frac{5}{147} a^{17} - \frac{3}{49} a^{16} - \frac{73}{147} a^{15} - \frac{10}{21} a^{14} + \frac{73}{147} a^{13} - \frac{1}{7} a^{12} - \frac{40}{147} a^{11} + \frac{4}{147} a^{10} + \frac{34}{147} a^{9} + \frac{9}{49} a^{8} - \frac{2}{147} a^{7} - \frac{41}{147} a^{6} + \frac{52}{147} a^{5} - \frac{64}{147} a^{4} + \frac{1}{7} a^{3} + \frac{3}{49} a^{2} - \frac{1}{7} a - \frac{1}{3}$, $\frac{1}{5562044819390785013051754702074243433} a^{19} - \frac{181005640568638262001926260800910}{5562044819390785013051754702074243433} a^{18} + \frac{38046243075102198358979888077518137}{1854014939796928337683918234024747811} a^{17} - \frac{672244504051558949281797153072997}{5562044819390785013051754702074243433} a^{16} - \frac{1698076127702804603904165503025261733}{5562044819390785013051754702074243433} a^{15} - \frac{1813555575321212443879186700068565837}{5562044819390785013051754702074243433} a^{14} + \frac{30183945964665844981163398917541184}{1854014939796928337683918234024747811} a^{13} - \frac{1074091366024807392352755027225206545}{5562044819390785013051754702074243433} a^{12} + \frac{1987668193205999633991095786740004242}{5562044819390785013051754702074243433} a^{11} - \frac{2774955935191596338869862208121245566}{5562044819390785013051754702074243433} a^{10} + \frac{20790530229966903521428444330104162}{109059702340995784569642249060279283} a^{9} + \frac{1619959932115794556017458938765031521}{5562044819390785013051754702074243433} a^{8} - \frac{1949608959841543022934333289356512063}{5562044819390785013051754702074243433} a^{7} + \frac{320450497484731064903031416327868145}{5562044819390785013051754702074243433} a^{6} - \frac{523934297567757238715874284456398555}{5562044819390785013051754702074243433} a^{5} + \frac{690748120471509244221809093856068182}{1854014939796928337683918234024747811} a^{4} + \frac{255521127154673022266449418762219718}{1854014939796928337683918234024747811} a^{3} + \frac{368240144394131317053611343538605491}{1854014939796928337683918234024747811} a^{2} - \frac{193808758199077760525832653282754703}{794577831341540716150250671724891919} a - \frac{14033902993611016248708482665001531}{37837039587692415054773841510709139}$

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:  \( 9290301067.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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 $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
109Data not computed