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Properties

Label 20.12.1853091914...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{7}$
Root discriminant $129.84$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5478679059725, 0, -919625677300, 0, -221238827475, 0, 42893457655, 0, -51334460, 0, -229221570, 0, 5882811, 0, 168742, 0, -5347, 0, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 23*x^18 - 5347*x^16 + 168742*x^14 + 5882811*x^12 - 229221570*x^10 - 51334460*x^8 + 42893457655*x^6 - 221238827475*x^4 - 919625677300*x^2 + 5478679059725)
 
gp: K = bnfinit(x^20 - 23*x^18 - 5347*x^16 + 168742*x^14 + 5882811*x^12 - 229221570*x^10 - 51334460*x^8 + 42893457655*x^6 - 221238827475*x^4 - 919625677300*x^2 + 5478679059725, 1)
 

Normalized defining polynomial

\( x^{20} - 23 x^{18} - 5347 x^{16} + 168742 x^{14} + 5882811 x^{12} - 229221570 x^{10} - 51334460 x^{8} + 42893457655 x^{6} - 221238827475 x^{4} - 919625677300 x^{2} + 5478679059725 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1853091914135141046229493177600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.84$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{12} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} - \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7}$, $\frac{1}{30145} a^{16} - \frac{23}{30145} a^{14} - \frac{5347}{30145} a^{12} - \frac{12128}{30145} a^{10} + \frac{4536}{30145} a^{8} + \frac{202}{6029} a^{6} + \frac{495}{6029} a^{4} - \frac{2714}{6029} a^{2}$, $\frac{1}{30145} a^{17} - \frac{23}{30145} a^{15} - \frac{5347}{30145} a^{13} - \frac{12128}{30145} a^{11} + \frac{4536}{30145} a^{9} + \frac{202}{6029} a^{7} + \frac{495}{6029} a^{5} - \frac{2714}{6029} a^{3}$, $\frac{1}{16807185086688930854670726744443863842760370918114657585} a^{18} - \frac{10598679119328295090826404858288766353365515590581}{16807185086688930854670726744443863842760370918114657585} a^{16} + \frac{21035444182202718061818232830435942879889248739504561}{1292860391283763911897748211111066449443105455239589045} a^{14} + \frac{756822346578950756133555233790010931182069338560958265}{3361437017337786170934145348888772768552074183622931517} a^{12} - \frac{2189508238614861041538718884259628619749843328578565937}{16807185086688930854670726744443863842760370918114657585} a^{10} + \frac{4489987146115082019811600838703042233078710323928410809}{16807185086688930854670726744443863842760370918114657585} a^{8} + \frac{3622857946555845221263989939169621977640286079299214711}{16807185086688930854670726744443863842760370918114657585} a^{6} - \frac{395255486982358772305535444397522471132010774252043232}{3361437017337786170934145348888772768552074183622931517} a^{4} + \frac{177227741740268114828268179408327071870208189374713}{557544703489432106640262953871085216213646406306673} a^{2} + \frac{12698143679374815500225922934156402344942016963}{92477144383717383751909595931511895208765368437}$, $\frac{1}{84035925433444654273353633722219319213801854590573287925} a^{19} + \frac{546946024370103811549436549012796449860280890716092}{84035925433444654273353633722219319213801854590573287925} a^{17} + \frac{537193175451073133709169359741090602659214979501105296}{6464301956418819559488741055555332247215527276197945225} a^{15} - \frac{2558516814001025864471855194287410763736094825339920723}{84035925433444654273353633722219319213801854590573287925} a^{13} + \frac{28024396788180954249403770848968350332083869075586350606}{84035925433444654273353633722219319213801854590573287925} a^{11} + \frac{4092951598098858147853682998603475169606421431485441121}{16807185086688930854670726744443863842760370918114657585} a^{9} + \frac{436354085270229759633557848814278543324808692676592699}{3361437017337786170934145348888772768552074183622931517} a^{7} - \frac{119270858755089879518605282231335289106255803130240097}{16807185086688930854670726744443863842760370918114657585} a^{5} - \frac{14751045623428172934882892789959242345276204112661}{557544703489432106640262953871085216213646406306673} a^{3} + \frac{21035057612618439850427103773133659510741477080}{92477144383717383751909595931511895208765368437} a$

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:  $15$
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:  \( 77244808806100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed