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Properties

Label 20.4.15495141543...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{16}\cdot 6329^{5}$
Root discriminant $32.32$
Ramified primes $5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-829, 1131, 3698, -1454, -9967, 21628, -7904, 1228, 5309, -351, -389, 19, 123, -159, -90, -20, 0, -2, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 - 2*x^17 - 20*x^15 - 90*x^14 - 159*x^13 + 123*x^12 + 19*x^11 - 389*x^10 - 351*x^9 + 5309*x^8 + 1228*x^7 - 7904*x^6 + 21628*x^5 - 9967*x^4 - 1454*x^3 + 3698*x^2 + 1131*x - 829)
 
gp: K = bnfinit(x^20 - x^18 - 2*x^17 - 20*x^15 - 90*x^14 - 159*x^13 + 123*x^12 + 19*x^11 - 389*x^10 - 351*x^9 + 5309*x^8 + 1228*x^7 - 7904*x^6 + 21628*x^5 - 9967*x^4 - 1454*x^3 + 3698*x^2 + 1131*x - 829, 1)
 

Normalized defining polynomial

\( x^{20} - x^{18} - 2 x^{17} - 20 x^{15} - 90 x^{14} - 159 x^{13} + 123 x^{12} + 19 x^{11} - 389 x^{10} - 351 x^{9} + 5309 x^{8} + 1228 x^{7} - 7904 x^{6} + 21628 x^{5} - 9967 x^{4} - 1454 x^{3} + 3698 x^{2} + 1131 x - 829 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1549514154367670081939697265625=5^{16}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.32$
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:  $5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{96774194727455376713596852171847030606289389} a^{19} + \frac{28332340717949109712346461629830212337175776}{96774194727455376713596852171847030606289389} a^{18} - \frac{664252651494512587210469490157352420722293}{96774194727455376713596852171847030606289389} a^{17} + \frac{9242051744765448159839126875056230589379965}{96774194727455376713596852171847030606289389} a^{16} + \frac{17459581897337922094590453125859575229453678}{96774194727455376713596852171847030606289389} a^{15} - \frac{22423461851518322367259136890941329992085228}{96774194727455376713596852171847030606289389} a^{14} - \frac{33961284868171814357002135713014443164177177}{96774194727455376713596852171847030606289389} a^{13} + \frac{23306832368650113795585056445829768710509222}{96774194727455376713596852171847030606289389} a^{12} + \frac{40713690767686913930392824153082618271189205}{96774194727455376713596852171847030606289389} a^{11} + \frac{6547284875718993850581924941697480829140787}{96774194727455376713596852171847030606289389} a^{10} - \frac{19935048321703977574256916655782676468816805}{96774194727455376713596852171847030606289389} a^{9} + \frac{39095956461047411875609337392120231313808566}{96774194727455376713596852171847030606289389} a^{8} + \frac{45649731664976516449332034758410603813764}{96774194727455376713596852171847030606289389} a^{7} + \frac{20376117579037534601021325578259344028272936}{96774194727455376713596852171847030606289389} a^{6} - \frac{33944683210500620677795313608393452671909235}{96774194727455376713596852171847030606289389} a^{5} - \frac{20928643433690248666842039117905069143153804}{96774194727455376713596852171847030606289389} a^{4} - \frac{29297144472896377871477764288390300414500639}{96774194727455376713596852171847030606289389} a^{3} + \frac{41829577640589309804854009107811687145433140}{96774194727455376713596852171847030606289389} a^{2} - \frac{24738301257402085236776659782997000135196108}{96774194727455376713596852171847030606289389} a - \frac{10155437627522091893148498715396541527753974}{96774194727455376713596852171847030606289389}$

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:  $11$
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:  \( 13650629.9215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
6329Data not computed