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Properties

Label 20.0.75570724234...3009.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{6}\cdot 397^{4}$
Root discriminant $19.68$
Ramified primes $3, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 16, -39, 82, -151, 253, -370, 439, -481, 502, -400, 259, -197, 152, -86, 35, -15, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 15*x^17 + 35*x^16 - 86*x^15 + 152*x^14 - 197*x^13 + 259*x^12 - 400*x^11 + 502*x^10 - 481*x^9 + 439*x^8 - 370*x^7 + 253*x^6 - 151*x^5 + 82*x^4 - 39*x^3 + 16*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 15*x^17 + 35*x^16 - 86*x^15 + 152*x^14 - 197*x^13 + 259*x^12 - 400*x^11 + 502*x^10 - 481*x^9 + 439*x^8 - 370*x^7 + 253*x^6 - 151*x^5 + 82*x^4 - 39*x^3 + 16*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 15 x^{17} + 35 x^{16} - 86 x^{15} + 152 x^{14} - 197 x^{13} + 259 x^{12} - 400 x^{11} + 502 x^{10} - 481 x^{9} + 439 x^{8} - 370 x^{7} + 253 x^{6} - 151 x^{5} + 82 x^{4} - 39 x^{3} + 16 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(75570724234611059753853009=3^{10}\cdot 61^{6}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.68$
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:  $3, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{6}{47} a^{17} - \frac{8}{47} a^{16} - \frac{4}{47} a^{15} - \frac{4}{47} a^{14} + \frac{14}{47} a^{13} - \frac{1}{47} a^{12} - \frac{15}{47} a^{11} + \frac{16}{47} a^{10} - \frac{9}{47} a^{9} - \frac{14}{47} a^{8} - \frac{16}{47} a^{7} - \frac{8}{47} a^{6} + \frac{22}{47} a^{5} - \frac{21}{47} a^{3} + \frac{9}{47} a^{2} - \frac{14}{47} a + \frac{2}{47}$, $\frac{1}{63595769513} a^{19} - \frac{567950219}{63595769513} a^{18} - \frac{344447991}{63595769513} a^{17} - \frac{8385511928}{63595769513} a^{16} - \frac{19785896945}{63595769513} a^{15} - \frac{27756236357}{63595769513} a^{14} - \frac{21188582310}{63595769513} a^{13} - \frac{9768483959}{63595769513} a^{12} + \frac{18416959711}{63595769513} a^{11} - \frac{30735080903}{63595769513} a^{10} + \frac{23586285818}{63595769513} a^{9} - \frac{3539204347}{63595769513} a^{8} + \frac{5622020394}{63595769513} a^{7} + \frac{21752403754}{63595769513} a^{6} + \frac{25247734345}{63595769513} a^{5} - \frac{21152846529}{63595769513} a^{4} + \frac{31027840915}{63595769513} a^{3} - \frac{18779328604}{63595769513} a^{2} + \frac{17543833549}{63595769513} a - \frac{25920760576}{63595769513}$

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:  \( -\frac{84792575767}{63595769513} a^{19} + \frac{386595207224}{63595769513} a^{18} - \frac{691874906188}{63595769513} a^{17} + \frac{1028983497358}{63595769513} a^{16} - \frac{2613740454217}{63595769513} a^{15} + \frac{6275620794228}{63595769513} a^{14} - \frac{10500618608809}{63595769513} a^{13} + \frac{12999359187047}{63595769513} a^{12} - \frac{17683888526951}{63595769513} a^{11} + \frac{27716295210272}{63595769513} a^{10} - \frac{32481926037842}{63595769513} a^{9} + \frac{30017056078580}{63595769513} a^{8} - \frac{27845611963166}{63595769513} a^{7} + \frac{22075905138974}{63595769513} a^{6} - \frac{14183408959450}{63595769513} a^{5} + \frac{8204566375769}{63595769513} a^{4} - \frac{4103657423832}{63595769513} a^{3} + \frac{1754272903548}{63595769513} a^{2} - \frac{668235019627}{63595769513} a + \frac{193664607176}{63595769513} \) (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:  \( 96029.8763499 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.24217.1, 10.0.142510530627.1, 10.8.8693142368247.1, 10.2.35774248429.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61Data not computed
397Data not computed