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Properties

Label 20.20.1252655737...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{24}\cdot 5^{10}\cdot 967^{8}$
Root discriminant $80.33$
Ramified primes $2, 5, 967$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6400, 0, -1740160, 0, 13155456, 0, -17142880, 0, 9338864, 0, -2669912, 0, 439496, 0, -43054, 0, 2475, 0, -77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 77*x^18 + 2475*x^16 - 43054*x^14 + 439496*x^12 - 2669912*x^10 + 9338864*x^8 - 17142880*x^6 + 13155456*x^4 - 1740160*x^2 + 6400)
 
gp: K = bnfinit(x^20 - 77*x^18 + 2475*x^16 - 43054*x^14 + 439496*x^12 - 2669912*x^10 + 9338864*x^8 - 17142880*x^6 + 13155456*x^4 - 1740160*x^2 + 6400, 1)
 

Normalized defining polynomial

\( x^{20} - 77 x^{18} + 2475 x^{16} - 43054 x^{14} + 439496 x^{12} - 2669912 x^{10} + 9338864 x^{8} - 17142880 x^{6} + 13155456 x^{4} - 1740160 x^{2} + 6400 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(125265573758493326836079165440000000000=2^{24}\cdot 5^{10}\cdot 967^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.33$
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:  $2, 5, 967$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{352} a^{14} + \frac{5}{352} a^{12} - \frac{19}{352} a^{10} - \frac{1}{8} a^{8} - \frac{5}{44} a^{6} - \frac{9}{44} a^{4} - \frac{9}{22} a^{2} - \frac{5}{11}$, $\frac{1}{352} a^{15} + \frac{5}{352} a^{13} - \frac{19}{352} a^{11} - \frac{1}{8} a^{9} - \frac{5}{44} a^{7} - \frac{9}{44} a^{5} - \frac{9}{22} a^{3} - \frac{5}{11} a$, $\frac{1}{3520} a^{16} - \frac{3}{3520} a^{14} + \frac{73}{3520} a^{12} - \frac{7}{220} a^{10} + \frac{67}{880} a^{8} - \frac{7}{88} a^{6} - \frac{3}{110} a^{4} - \frac{1}{2} a^{3} + \frac{53}{110} a^{2} + \frac{4}{11}$, $\frac{1}{3520} a^{17} - \frac{3}{3520} a^{15} + \frac{73}{3520} a^{13} - \frac{7}{220} a^{11} + \frac{67}{880} a^{9} - \frac{7}{88} a^{7} - \frac{3}{110} a^{5} - \frac{1}{55} a^{3} - \frac{1}{2} a^{2} + \frac{4}{11} a$, $\frac{1}{158675069832583040} a^{18} + \frac{6655939612583}{158675069832583040} a^{16} - \frac{761432970269}{31735013966516608} a^{14} - \frac{2133259031191637}{79337534916291520} a^{12} + \frac{106558391816307}{1803125793552080} a^{10} + \frac{1350309864678091}{19834383729072880} a^{8} + \frac{196375873303188}{1239648983067055} a^{6} - \frac{34749894930851}{495859593226822} a^{4} - \frac{1}{2} a^{3} + \frac{307128261468469}{2479297966134110} a^{2} - \frac{1484876114608}{247929796613411}$, $\frac{1}{793375349162915200} a^{19} + \frac{6655939612583}{793375349162915200} a^{17} + \frac{17878971341467}{31735013966516608} a^{15} - \frac{5964901342489807}{396687674581457600} a^{13} - \frac{7234821158286671}{198343837290728800} a^{11} + \frac{2506963186536869}{24792979661341100} a^{9} - \frac{8233489516013931}{49585959322682200} a^{7} - \frac{254035207869136}{1239648983067055} a^{5} - \frac{353564998702288}{6198244915335275} a^{3} + \frac{133749558401798}{1239648983067055} a$

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:  $19$
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:  \( 25566000123300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.4675445.1, 10.10.11192210405388800000.1, 10.10.89537683243110400.1, 10.10.2732473243503125.1

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 30 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
2.8.12.8$x^{8} + 2 x^{6} + 80$$2$$4$$12$$(C_8:C_2):C_2$$[2, 2, 3]^{4}$
2.8.12.8$x^{8} + 2 x^{6} + 80$$2$$4$$12$$(C_8:C_2):C_2$$[2, 2, 3]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
967Data not computed