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Properties

Label 20.4.13878788557...1232.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 17^{17}$
Root discriminant $25.53$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_4\times A_5$ (as 20T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -60, 340, -1080, 2298, -3954, 6050, -7370, 5743, -1857, -1088, 1405, -435, -196, 146, -24, 5, 19, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 8*x^18 + 19*x^17 + 5*x^16 - 24*x^15 + 146*x^14 - 196*x^13 - 435*x^12 + 1405*x^11 - 1088*x^10 - 1857*x^9 + 5743*x^8 - 7370*x^7 + 6050*x^6 - 3954*x^5 + 2298*x^4 - 1080*x^3 + 340*x^2 - 60*x + 4)
 
gp: K = bnfinit(x^20 - 3*x^19 - 8*x^18 + 19*x^17 + 5*x^16 - 24*x^15 + 146*x^14 - 196*x^13 - 435*x^12 + 1405*x^11 - 1088*x^10 - 1857*x^9 + 5743*x^8 - 7370*x^7 + 6050*x^6 - 3954*x^5 + 2298*x^4 - 1080*x^3 + 340*x^2 - 60*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 8 x^{18} + 19 x^{17} + 5 x^{16} - 24 x^{15} + 146 x^{14} - 196 x^{13} - 435 x^{12} + 1405 x^{11} - 1088 x^{10} - 1857 x^{9} + 5743 x^{8} - 7370 x^{7} + 6050 x^{6} - 3954 x^{5} + 2298 x^{4} - 1080 x^{3} + 340 x^{2} - 60 x + 4 \)

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:  \(13878788557563639341338591232=2^{24}\cdot 17^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.53$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{3}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{3}{16} a^{12} - \frac{3}{16} a^{11} - \frac{3}{16} a^{10} - \frac{1}{4} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{32} a^{18} - \frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{8} a^{14} + \frac{3}{16} a^{12} + \frac{3}{32} a^{10} + \frac{3}{32} a^{9} - \frac{7}{32} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{21417507813383751498944} a^{19} - \frac{16254244323376516147}{1338594238336484468684} a^{18} - \frac{323177933673559466113}{10708753906691875749472} a^{17} + \frac{316181019182454722335}{21417507813383751498944} a^{16} + \frac{196203537539097041121}{5354376953345937874736} a^{15} + \frac{258337511352224704109}{5354376953345937874736} a^{14} + \frac{845047259443788777103}{10708753906691875749472} a^{13} - \frac{1885521527410011832029}{10708753906691875749472} a^{12} + \frac{3130427561058797771003}{21417507813383751498944} a^{11} - \frac{130167418018225089713}{10708753906691875749472} a^{10} - \frac{1156658971862148120579}{5354376953345937874736} a^{9} - \frac{4977769433626346595163}{21417507813383751498944} a^{8} + \frac{1141187977510064434585}{5354376953345937874736} a^{7} + \frac{49042159800134080749}{10708753906691875749472} a^{6} + \frac{477489436317473061689}{1338594238336484468684} a^{5} + \frac{2826560233341025272221}{10708753906691875749472} a^{4} + \frac{646882610038968601495}{5354376953345937874736} a^{3} - \frac{369711363953656791351}{5354376953345937874736} a^{2} - \frac{3047738991273137589}{669297119168242234342} a - \frac{968277192275928095497}{5354376953345937874736}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3329080.48178 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 5.1.18496.1, 10.2.1680747204608.1

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

Sibling fields

Degree 24 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 $20$ $20$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
17Data not computed