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Properties

Label 20.4.49338146756...1664.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 11^{16}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -28, -108, -21, 410, 154, -880, -509, 1012, 562, -792, -328, 440, 88, -162, 5, 28, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 2*x^18 + 28*x^17 + 5*x^16 - 162*x^15 + 88*x^14 + 440*x^13 - 328*x^12 - 792*x^11 + 562*x^10 + 1012*x^9 - 509*x^8 - 880*x^7 + 154*x^6 + 410*x^5 - 21*x^4 - 108*x^3 - 28*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 2*x^18 + 28*x^17 + 5*x^16 - 162*x^15 + 88*x^14 + 440*x^13 - 328*x^12 - 792*x^11 + 562*x^10 + 1012*x^9 - 509*x^8 - 880*x^7 + 154*x^6 + 410*x^5 - 21*x^4 - 108*x^3 - 28*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 2 x^{18} + 28 x^{17} + 5 x^{16} - 162 x^{15} + 88 x^{14} + 440 x^{13} - 328 x^{12} - 792 x^{11} + 562 x^{10} + 1012 x^{9} - 509 x^{8} - 880 x^{7} + 154 x^{6} + 410 x^{5} - 21 x^{4} - 108 x^{3} - 28 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49338146756019243307761664=2^{30}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.26$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{3}{11} a^{14} - \frac{5}{11} a^{13} + \frac{3}{11} a^{12} - \frac{2}{11} a^{11} - \frac{4}{11} a^{10} - \frac{2}{11} a^{8} + \frac{5}{11} a^{7} - \frac{4}{11} a^{5} + \frac{4}{11} a^{4} + \frac{1}{11} a^{3} - \frac{4}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{3}{11} a^{14} - \frac{4}{11} a^{13} + \frac{2}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} + \frac{5}{11} a^{5} + \frac{4}{11} a^{3} + \frac{5}{11} a^{2} - \frac{3}{11}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} - \frac{4}{11} a^{13} - \frac{5}{11} a^{11} - \frac{3}{11} a^{10} + \frac{1}{11} a^{8} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a^{2} - \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{14} + \frac{3}{11} a^{13} + \frac{2}{11} a^{12} - \frac{4}{11} a^{11} - \frac{2}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} + \frac{2}{11} a^{7} - \frac{1}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a^{2} + \frac{5}{11} a - \frac{5}{11}$, $\frac{1}{87416049234727631} a^{19} - \frac{2667728896724800}{87416049234727631} a^{18} - \frac{93402672049118}{87416049234727631} a^{17} - \frac{1266495396852374}{87416049234727631} a^{16} + \frac{207554896067996}{87416049234727631} a^{15} - \frac{2570871411064449}{87416049234727631} a^{14} - \frac{10723477228573401}{87416049234727631} a^{13} - \frac{41993927976960029}{87416049234727631} a^{12} + \frac{8203006929002834}{87416049234727631} a^{11} + \frac{39118044764115979}{87416049234727631} a^{10} + \frac{18675872914789162}{87416049234727631} a^{9} + \frac{32161520429052711}{87416049234727631} a^{8} - \frac{27154105829101431}{87416049234727631} a^{7} + \frac{23767448835131849}{87416049234727631} a^{6} - \frac{42256035021354267}{87416049234727631} a^{5} - \frac{37005856028272383}{87416049234727631} a^{4} + \frac{36725798336339671}{87416049234727631} a^{3} - \frac{12499677837514948}{87416049234727631} a^{2} - \frac{35687369767799461}{87416049234727631} a + \frac{40052028521112986}{87416049234727631}$

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:  \( 94333.9687476 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5$ (as 20T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1 x2, 10.2.219503494144.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$