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Properties

Label 20.0.95012027507...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 19^{10}$
Root discriminant $62.93$
Ramified primes $2, 3, 5, 19$
Class number $20$ (GRH)
Class group $[2, 10]$ (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![11335609, 13635882, 8853151, 13098912, 10244121, 4501338, 6060696, 1568340, 1997289, 362082, 543153, 28992, 107613, -648, 13614, -228, 1077, -12, 49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 49*x^18 - 12*x^17 + 1077*x^16 - 228*x^15 + 13614*x^14 - 648*x^13 + 107613*x^12 + 28992*x^11 + 543153*x^10 + 362082*x^9 + 1997289*x^8 + 1568340*x^7 + 6060696*x^6 + 4501338*x^5 + 10244121*x^4 + 13098912*x^3 + 8853151*x^2 + 13635882*x + 11335609)
 
gp: K = bnfinit(x^20 + 49*x^18 - 12*x^17 + 1077*x^16 - 228*x^15 + 13614*x^14 - 648*x^13 + 107613*x^12 + 28992*x^11 + 543153*x^10 + 362082*x^9 + 1997289*x^8 + 1568340*x^7 + 6060696*x^6 + 4501338*x^5 + 10244121*x^4 + 13098912*x^3 + 8853151*x^2 + 13635882*x + 11335609, 1)
 

Normalized defining polynomial

\( x^{20} + 49 x^{18} - 12 x^{17} + 1077 x^{16} - 228 x^{15} + 13614 x^{14} - 648 x^{13} + 107613 x^{12} + 28992 x^{11} + 543153 x^{10} + 362082 x^{9} + 1997289 x^{8} + 1568340 x^{7} + 6060696 x^{6} + 4501338 x^{5} + 10244121 x^{4} + 13098912 x^{3} + 8853151 x^{2} + 13635882 x + 11335609 \)

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:  \(950120275075465393875600000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.93$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{18} a^{8} + \frac{4}{9} a^{6} - \frac{1}{2} a^{4} + \frac{7}{18} a^{2} - \frac{1}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{13} - \frac{1}{18} a^{9} + \frac{4}{9} a^{7} - \frac{1}{2} a^{5} + \frac{7}{18} a^{3} - \frac{1}{9} a$, $\frac{1}{36} a^{16} + \frac{1}{18} a^{12} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{2}{9} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{17}{36}$, $\frac{1}{36} a^{17} + \frac{1}{18} a^{13} + \frac{1}{18} a^{11} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{4}{9} a^{7} - \frac{1}{6} a^{6} + \frac{5}{18} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{13}{36} a - \frac{1}{6}$, $\frac{1}{36} a^{18} - \frac{1}{12} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{5}{18}$, $\frac{1}{52796470324535015232568723078321033001848918407894977916} a^{19} - \frac{31717766415430213688467335221074153626696890651586599}{52796470324535015232568723078321033001848918407894977916} a^{18} + \frac{640244491496763325531347639968039247855188443400179259}{52796470324535015232568723078321033001848918407894977916} a^{17} - \frac{70069824938779712662330421810525875372293883927118991}{5866274480503890581396524786480114777983213156432775324} a^{16} + \frac{108504941026480305918829945110207544499339363776092505}{4399705860377917936047393589860086083487409867324581493} a^{15} + \frac{26003852033931307883093483711154814226997784369797976}{13199117581133753808142180769580258250462229601973744479} a^{14} + \frac{71548316889024850052919683840798559895946225180844097}{13199117581133753808142180769580258250462229601973744479} a^{13} - \frac{950429210143020115077127306772017104587387532851155106}{13199117581133753808142180769580258250462229601973744479} a^{12} - \frac{218440037563893416512414122726071538936138866001799237}{3105674724972647954856983710489472529520524612229116348} a^{11} - \frac{564929371692343793974214744480381930250288307777081723}{17598823441511671744189574359440344333949639469298325972} a^{10} - \frac{2247831497116354588374128213916766943670362449158794333}{17598823441511671744189574359440344333949639469298325972} a^{9} - \frac{106042739668840857224871970728538936697800618261135541}{894855429229407037840147848785102254268625735727033524} a^{8} + \frac{8119072723303719051047505467075550585461835351788444167}{26398235162267507616284361539160516500924459203947488958} a^{7} + \frac{4429449570158590108505901788805206089017470719306209026}{13199117581133753808142180769580258250462229601973744479} a^{6} - \frac{3534445026857316828896841378525200453110258998962006659}{13199117581133753808142180769580258250462229601973744479} a^{5} - \frac{2107505088301613584635968952453134191884486795675718102}{4399705860377917936047393589860086083487409867324581493} a^{4} + \frac{1830384370545140555283929937712246457524611350312845581}{5866274480503890581396524786480114777983213156432775324} a^{3} - \frac{2986418973049807158082261646355371924304109317707436725}{52796470324535015232568723078321033001848918407894977916} a^{2} - \frac{7582030354886580361572201186919019486134877944691681377}{52796470324535015232568723078321033001848918407894977916} a + \frac{19522200300429657442979148426067062321418446494972464007}{52796470324535015232568723078321033001848918407894977916}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 809311494.1237729 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{285}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}, \sqrt{-19})\), 5.1.162000.1, 10.0.393660000000.1, 10.2.974741132340000000.1, 10.0.64982742156000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{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}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$