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Properties

Label 20.0.49338146756...1664.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
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![352, -704, 1760, -4576, 8976, -14048, 18704, -21328, 21128, -18560, 14560, -10176, 6408, -3640, 1840, -836, 338, -116, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352)
 
gp: K = bnfinit(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 34 x^{18} - 116 x^{17} + 338 x^{16} - 836 x^{15} + 1840 x^{14} - 3640 x^{13} + 6408 x^{12} - 10176 x^{11} + 14560 x^{10} - 18560 x^{9} + 21128 x^{8} - 21328 x^{7} + 18704 x^{6} - 14048 x^{5} + 8976 x^{4} - 4576 x^{3} + 1760 x^{2} - 704 x + 352 \)

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:  \(49338146756019243307761664=2^{30}\cdot 11^{16}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{688} a^{18} + \frac{19}{688} a^{17} + \frac{7}{344} a^{16} - \frac{9}{344} a^{15} - \frac{21}{344} a^{14} - \frac{15}{344} a^{13} + \frac{3}{86} a^{12} + \frac{3}{172} a^{11} - \frac{5}{86} a^{10} + \frac{4}{43} a^{9} - \frac{15}{172} a^{8} - \frac{7}{43} a^{7} + \frac{2}{43} a^{6} + \frac{1}{43} a^{5} + \frac{1}{43} a^{4} - \frac{8}{43} a^{3} - \frac{16}{43} a^{2} + \frac{20}{43} a + \frac{19}{43}$, $\frac{1}{3406356112} a^{19} - \frac{625525}{1703178056} a^{18} + \frac{29140343}{3406356112} a^{17} + \frac{39878045}{1703178056} a^{16} + \frac{5102225}{212897257} a^{15} - \frac{23406541}{1703178056} a^{14} - \frac{1161701}{1703178056} a^{13} + \frac{5472279}{851589028} a^{12} + \frac{36719243}{851589028} a^{11} - \frac{73021841}{851589028} a^{10} - \frac{15395449}{425794514} a^{9} - \frac{30789639}{425794514} a^{8} + \frac{6041046}{212897257} a^{7} - \frac{46713705}{212897257} a^{6} + \frac{64208451}{425794514} a^{5} - \frac{2045967}{425794514} a^{4} - \frac{9305043}{212897257} a^{3} - \frac{14462082}{212897257} a^{2} + \frac{14495725}{212897257} a - \frac{53617858}{212897257}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 43044.7776734 \)
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

\(\Q(\zeta_{11})^+\), 10.2.219503494144.1 x2, 10.6.219503494144.2

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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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}$