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Properties

Label 20.8.15418170861...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-131, 88, 1424, 70, -4024, -1018, 5348, 852, -2493, -398, -620, 790, 490, -414, -20, 98, -50, -6, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 20*x^18 - 6*x^17 - 50*x^16 + 98*x^15 - 20*x^14 - 414*x^13 + 490*x^12 + 790*x^11 - 620*x^10 - 398*x^9 - 2493*x^8 + 852*x^7 + 5348*x^6 - 1018*x^5 - 4024*x^4 + 70*x^3 + 1424*x^2 + 88*x - 131)
 
gp: K = bnfinit(x^20 - 8*x^19 + 20*x^18 - 6*x^17 - 50*x^16 + 98*x^15 - 20*x^14 - 414*x^13 + 490*x^12 + 790*x^11 - 620*x^10 - 398*x^9 - 2493*x^8 + 852*x^7 + 5348*x^6 - 1018*x^5 - 4024*x^4 + 70*x^3 + 1424*x^2 + 88*x - 131, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 20 x^{18} - 6 x^{17} - 50 x^{16} + 98 x^{15} - 20 x^{14} - 414 x^{13} + 490 x^{12} + 790 x^{11} - 620 x^{10} - 398 x^{9} - 2493 x^{8} + 852 x^{7} + 5348 x^{6} - 1018 x^{5} - 4024 x^{4} + 70 x^{3} + 1424 x^{2} + 88 x - 131 \)

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:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{906160164774708558183390967057} a^{19} - \frac{215566324804339251373596416540}{906160164774708558183390967057} a^{18} - \frac{438231883661925350358184283228}{906160164774708558183390967057} a^{17} - \frac{8565087517744044775699215517}{906160164774708558183390967057} a^{16} + \frac{69940447104964245092291192108}{906160164774708558183390967057} a^{15} - \frac{125389279917410478761407259415}{906160164774708558183390967057} a^{14} + \frac{222233311750432476711970141129}{906160164774708558183390967057} a^{13} + \frac{174626409078218385596039219940}{906160164774708558183390967057} a^{12} + \frac{45501020427727302601651188012}{906160164774708558183390967057} a^{11} - \frac{345322560791792419900809335867}{906160164774708558183390967057} a^{10} - \frac{80337456993758280527871379539}{906160164774708558183390967057} a^{9} + \frac{31958897053356484411025971606}{906160164774708558183390967057} a^{8} - \frac{72611491338893435099760314135}{906160164774708558183390967057} a^{7} - \frac{439435247353124999447765269525}{906160164774708558183390967057} a^{6} - \frac{158331733459278443380330842521}{906160164774708558183390967057} a^{5} + \frac{182276891465571346538627645825}{906160164774708558183390967057} a^{4} + \frac{238275809855569245707030845263}{906160164774708558183390967057} a^{3} + \frac{3688898544044439101703195491}{906160164774708558183390967057} a^{2} + \frac{17617588842516178634578142553}{39398268033682980790582215959} a - \frac{77660389082783723134221621011}{906160164774708558183390967057}$

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:  \( 11179336.4054 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.1

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 $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$