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Properties

Label 20.0.60915010078...9841.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 331^{2}$
Root discriminant $15.46$
Ramified primes $11, 331$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 110, -436, 1168, -2365, 3943, -5580, 6798, -7262, 6874, -5781, 4334, -2897, 1714, -891, 402, -151, 44, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 44*x^18 - 151*x^17 + 402*x^16 - 891*x^15 + 1714*x^14 - 2897*x^13 + 4334*x^12 - 5781*x^11 + 6874*x^10 - 7262*x^9 + 6798*x^8 - 5580*x^7 + 3943*x^6 - 2365*x^5 + 1168*x^4 - 436*x^3 + 110*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 44*x^18 - 151*x^17 + 402*x^16 - 891*x^15 + 1714*x^14 - 2897*x^13 + 4334*x^12 - 5781*x^11 + 6874*x^10 - 7262*x^9 + 6798*x^8 - 5580*x^7 + 3943*x^6 - 2365*x^5 + 1168*x^4 - 436*x^3 + 110*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 44 x^{18} - 151 x^{17} + 402 x^{16} - 891 x^{15} + 1714 x^{14} - 2897 x^{13} + 4334 x^{12} - 5781 x^{11} + 6874 x^{10} - 7262 x^{9} + 6798 x^{8} - 5580 x^{7} + 3943 x^{6} - 2365 x^{5} + 1168 x^{4} - 436 x^{3} + 110 x^{2} - 16 x + 1 \)

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:  \(609150100783522373289841=11^{18}\cdot 331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.46$
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:  $11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} - \frac{3}{89} a^{17} + \frac{17}{89} a^{16} - \frac{22}{89} a^{15} + \frac{28}{89} a^{14} + \frac{9}{89} a^{13} + \frac{3}{89} a^{12} - \frac{23}{89} a^{11} - \frac{14}{89} a^{10} + \frac{38}{89} a^{9} + \frac{19}{89} a^{8} - \frac{14}{89} a^{7} - \frac{43}{89} a^{6} - \frac{16}{89} a^{5} - \frac{38}{89} a^{4} + \frac{43}{89} a^{3} - \frac{12}{89} a^{2} - \frac{5}{89} a + \frac{10}{89}$, $\frac{1}{1174316076473} a^{19} - \frac{2723049969}{1174316076473} a^{18} - \frac{507251159711}{1174316076473} a^{17} - \frac{301090537734}{1174316076473} a^{16} - \frac{476797650147}{1174316076473} a^{15} - \frac{58230803663}{1174316076473} a^{14} - \frac{358761609059}{1174316076473} a^{13} + \frac{108975730191}{1174316076473} a^{12} - \frac{307130121305}{1174316076473} a^{11} - \frac{298112201976}{1174316076473} a^{10} + \frac{437769990031}{1174316076473} a^{9} - \frac{173113282856}{1174316076473} a^{8} + \frac{145759275184}{1174316076473} a^{7} - \frac{368283269739}{1174316076473} a^{6} - \frac{498563275296}{1174316076473} a^{5} + \frac{249465308890}{1174316076473} a^{4} + \frac{408478290049}{1174316076473} a^{3} + \frac{274056358089}{1174316076473} a^{2} - \frac{584376979517}{1174316076473} a + \frac{559917097408}{1174316076473}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{2262341817052}{1174316076473} a^{19} - \frac{19238631458105}{1174316076473} a^{18} + \frac{89843679674538}{1174316076473} a^{17} - \frac{295712540486335}{1174316076473} a^{16} + \frac{756513160801002}{1174316076473} a^{15} - \frac{1619734381604511}{1174316076473} a^{14} + \frac{3020707784286304}{1174316076473} a^{13} - \frac{4940180373760608}{1174316076473} a^{12} + \frac{7137540258720222}{1174316076473} a^{11} - \frac{9180620688835956}{1174316076473} a^{10} + \frac{10476682414332619}{1174316076473} a^{9} - \frac{10556558175495978}{1174316076473} a^{8} + \frac{9364826780195900}{1174316076473} a^{7} - \frac{7187023385803427}{1174316076473} a^{6} + \frac{4646500118670624}{1174316076473} a^{5} - \frac{2496444059821590}{1174316076473} a^{4} + \frac{1046555416140498}{1174316076473} a^{3} - \frac{274367121327883}{1174316076473} a^{2} + \frac{32383954584233}{1174316076473} a + \frac{200766432524}{1174316076473} \) (order $22$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 27553.4171511 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\), 10.8.70952789611.1, 10.2.780480685721.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
331Data not computed