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Properties

Label 20.0.15551992186...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{16}\cdot 269^{7}$
Root discriminant $25.68$
Ramified primes $5, 269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![269, -649, 5691, -29086, 86332, -142143, 182542, -177441, 147375, -103368, 65251, -36406, 18790, -8748, 3797, -1476, 523, -158, 41, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 41*x^18 - 158*x^17 + 523*x^16 - 1476*x^15 + 3797*x^14 - 8748*x^13 + 18790*x^12 - 36406*x^11 + 65251*x^10 - 103368*x^9 + 147375*x^8 - 177441*x^7 + 182542*x^6 - 142143*x^5 + 86332*x^4 - 29086*x^3 + 5691*x^2 - 649*x + 269)
 
gp: K = bnfinit(x^20 - 8*x^19 + 41*x^18 - 158*x^17 + 523*x^16 - 1476*x^15 + 3797*x^14 - 8748*x^13 + 18790*x^12 - 36406*x^11 + 65251*x^10 - 103368*x^9 + 147375*x^8 - 177441*x^7 + 182542*x^6 - 142143*x^5 + 86332*x^4 - 29086*x^3 + 5691*x^2 - 649*x + 269, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 41 x^{18} - 158 x^{17} + 523 x^{16} - 1476 x^{15} + 3797 x^{14} - 8748 x^{13} + 18790 x^{12} - 36406 x^{11} + 65251 x^{10} - 103368 x^{9} + 147375 x^{8} - 177441 x^{7} + 182542 x^{6} - 142143 x^{5} + 86332 x^{4} - 29086 x^{3} + 5691 x^{2} - 649 x + 269 \)

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:  \(15551992186695249786376953125=5^{16}\cdot 269^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $5, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{17} - \frac{4}{25} a^{16} + \frac{7}{25} a^{15} - \frac{8}{25} a^{14} + \frac{9}{25} a^{13} - \frac{2}{5} a^{12} - \frac{4}{25} a^{11} + \frac{4}{25} a^{10} - \frac{9}{25} a^{9} - \frac{1}{25} a^{8} - \frac{6}{25} a^{7} + \frac{3}{25} a^{5} + \frac{4}{25} a^{4} + \frac{1}{25} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{25}$, $\frac{1}{7361395882856287849725483474617511163925} a^{19} + \frac{113606252986854291228164025184131011042}{7361395882856287849725483474617511163925} a^{18} - \frac{388158320030994812407906941648388960478}{7361395882856287849725483474617511163925} a^{17} + \frac{134598388261416825820179366986048472123}{7361395882856287849725483474617511163925} a^{16} + \frac{1129409359700823535978926624874819460224}{7361395882856287849725483474617511163925} a^{15} - \frac{1616461681018191167674127410794446240934}{7361395882856287849725483474617511163925} a^{14} - \frac{449632154499674675939279502661278210926}{7361395882856287849725483474617511163925} a^{13} + \frac{2442735734970269559727322424751772900556}{7361395882856287849725483474617511163925} a^{12} - \frac{47776087987948296976244456498676277174}{1472279176571257569945096694923502232785} a^{11} - \frac{409151075561152990303761299924708285636}{1472279176571257569945096694923502232785} a^{10} + \frac{3914715320162792340490751176386134807}{294455835314251513989019338984700446557} a^{9} + \frac{1375034107912073384340741703108187474353}{7361395882856287849725483474617511163925} a^{8} + \frac{2620642737822651467560207799169772837069}{7361395882856287849725483474617511163925} a^{7} - \frac{102862216381070988331525254556343634077}{7361395882856287849725483474617511163925} a^{6} - \frac{276114920933834050042223430259654720033}{7361395882856287849725483474617511163925} a^{5} + \frac{75418382669786184210039099559692121717}{294455835314251513989019338984700446557} a^{4} + \frac{387539607803227508957946067555140108606}{7361395882856287849725483474617511163925} a^{3} - \frac{712109801642258988990812581200342912211}{1472279176571257569945096694923502232785} a^{2} - \frac{1028973651271254262023431235341111558754}{7361395882856287849725483474617511163925} a - \frac{2494852945293760800788610608159091230114}{7361395882856287849725483474617511163925}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 696572.370086 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.33625.1, 10.2.5653203125.1

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

Sibling fields

Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
269Data not computed