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Properties

Label 20.0.31874096343...8949.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{19}\cdot 223^{7}$
Root discriminant $18.84$
Ramified primes $3, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -434, 268, 696, -396, -1107, 1050, 222, -1023, 973, -377, -50, 153, -219, 243, -180, 108, -48, 19, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 19*x^18 - 48*x^17 + 108*x^16 - 180*x^15 + 243*x^14 - 219*x^13 + 153*x^12 - 50*x^11 - 377*x^10 + 973*x^9 - 1023*x^8 + 222*x^7 + 1050*x^6 - 1107*x^5 - 396*x^4 + 696*x^3 + 268*x^2 - 434*x + 109)
 
gp: K = bnfinit(x^20 - 5*x^19 + 19*x^18 - 48*x^17 + 108*x^16 - 180*x^15 + 243*x^14 - 219*x^13 + 153*x^12 - 50*x^11 - 377*x^10 + 973*x^9 - 1023*x^8 + 222*x^7 + 1050*x^6 - 1107*x^5 - 396*x^4 + 696*x^3 + 268*x^2 - 434*x + 109, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 19 x^{18} - 48 x^{17} + 108 x^{16} - 180 x^{15} + 243 x^{14} - 219 x^{13} + 153 x^{12} - 50 x^{11} - 377 x^{10} + 973 x^{9} - 1023 x^{8} + 222 x^{7} + 1050 x^{6} - 1107 x^{5} - 396 x^{4} + 696 x^{3} + 268 x^{2} - 434 x + 109 \)

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:  \(31874096343147504926018949=3^{19}\cdot 223^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.84$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{51004073074042978016178699050245} a^{19} - \frac{12221154417397138923366321648122}{51004073074042978016178699050245} a^{18} + \frac{3769113088224816067876856575163}{51004073074042978016178699050245} a^{17} - \frac{18569248204258831235953221956969}{51004073074042978016178699050245} a^{16} + \frac{7778468352956210066515904713151}{51004073074042978016178699050245} a^{15} - \frac{16910391076527568720347428934977}{51004073074042978016178699050245} a^{14} - \frac{19536031585486945319998327507198}{51004073074042978016178699050245} a^{13} + \frac{15842515311231235452297657542747}{51004073074042978016178699050245} a^{12} - \frac{21300941228863432136701568056401}{51004073074042978016178699050245} a^{11} + \frac{725370373168849012758279047632}{51004073074042978016178699050245} a^{10} + \frac{24377716934924060550574945496454}{51004073074042978016178699050245} a^{9} + \frac{2132449858332554037603339934002}{10200814614808595603235739810049} a^{8} - \frac{11798368240401287155087476033493}{51004073074042978016178699050245} a^{7} - \frac{8864142714366465783867242889352}{51004073074042978016178699050245} a^{6} + \frac{21129906262470257066486268412474}{51004073074042978016178699050245} a^{5} + \frac{1958633727813720901043804165390}{10200814614808595603235739810049} a^{4} + \frac{25425156710889919736589826438749}{51004073074042978016178699050245} a^{3} + \frac{19925339210760587304826781339318}{51004073074042978016178699050245} a^{2} + \frac{25489318969736366927568453931192}{51004073074042978016178699050245} a - \frac{1263625772716180945283284352118}{51004073074042978016178699050245}$

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:  \( \frac{89823302447239604}{1619643228348147295} a^{19} - \frac{388841908651209123}{1619643228348147295} a^{18} + \frac{1451743123667227272}{1619643228348147295} a^{17} - \frac{3357892374386216281}{1619643228348147295} a^{16} + \frac{7525345587834591984}{1619643228348147295} a^{15} - \frac{11267788976331523453}{1619643228348147295} a^{14} + \frac{14612510788172315083}{1619643228348147295} a^{13} - \frac{10238623224133236752}{1619643228348147295} a^{12} + \frac{7331741760310562436}{1619643228348147295} a^{11} + \frac{542616985514655108}{1619643228348147295} a^{10} - \frac{33629969905951738609}{1619643228348147295} a^{9} + \frac{13170687806005616598}{323928645669629459} a^{8} - \frac{50549810693992731302}{1619643228348147295} a^{7} - \frac{10522920150829056498}{1619643228348147295} a^{6} + \frac{85571744400754714666}{1619643228348147295} a^{5} - \frac{8737624030130584640}{323928645669629459} a^{4} - \frac{59673395851879241779}{1619643228348147295} a^{3} + \frac{21652529147766568362}{1619643228348147295} a^{2} + \frac{36239922324048387853}{1619643228348147295} a - \frac{15277414532064744732}{1619643228348147295} \) (order $6$)
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:  \( 119850.009202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.6021.1, 5.3.18063.1, 10.0.978815907.1

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 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 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
223Data not computed