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Properties

Label 20.16.2821193160...1264.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{40}\cdot 11^{16}\cdot 89^{5}$
Root discriminant $83.66$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![399521, 0, -756286, 0, 252478, 0, 264880, 0, -197479, 0, 26484, 0, 11775, 0, -4664, 0, 659, 0, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 42*x^18 + 659*x^16 - 4664*x^14 + 11775*x^12 + 26484*x^10 - 197479*x^8 + 264880*x^6 + 252478*x^4 - 756286*x^2 + 399521)
 
gp: K = bnfinit(x^20 - 42*x^18 + 659*x^16 - 4664*x^14 + 11775*x^12 + 26484*x^10 - 197479*x^8 + 264880*x^6 + 252478*x^4 - 756286*x^2 + 399521, 1)
 

Normalized defining polynomial

\( x^{20} - 42 x^{18} + 659 x^{16} - 4664 x^{14} + 11775 x^{12} + 26484 x^{10} - 197479 x^{8} + 264880 x^{6} + 252478 x^{4} - 756286 x^{2} + 399521 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(282119316059236304095781411736743051264=2^{40}\cdot 11^{16}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.66$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{2}{9} a^{10} - \frac{4}{9} a^{8} - \frac{1}{3} a^{6} - \frac{4}{9} a^{4} + \frac{4}{9} a^{2} + \frac{4}{9}$, $\frac{1}{9} a^{15} + \frac{2}{9} a^{11} - \frac{4}{9} a^{9} - \frac{1}{3} a^{7} - \frac{4}{9} a^{5} + \frac{4}{9} a^{3} + \frac{4}{9} a$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} + \frac{2}{27} a^{12} + \frac{1}{9} a^{10} + \frac{1}{27} a^{8} - \frac{1}{27} a^{6} - \frac{10}{27} a^{4} + \frac{5}{27}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{15} + \frac{2}{27} a^{13} + \frac{1}{9} a^{11} + \frac{1}{27} a^{9} - \frac{1}{27} a^{7} - \frac{10}{27} a^{5} + \frac{5}{27} a$, $\frac{1}{197345169991461354669} a^{18} + \frac{2089618003003529572}{197345169991461354669} a^{16} + \frac{2399514231524804}{2860074927412483401} a^{14} + \frac{2413714017617483905}{197345169991461354669} a^{12} + \frac{40920728141001050314}{197345169991461354669} a^{10} + \frac{89047158074437731505}{197345169991461354669} a^{8} - \frac{4670293756800501458}{21927241110162372741} a^{6} + \frac{4827009580991589979}{197345169991461354669} a^{4} + \frac{8452977094067409797}{197345169991461354669} a^{2} + \frac{57250604172383448466}{197345169991461354669}$, $\frac{1}{13222126389427910762823} a^{19} - \frac{158710150138187203862}{13222126389427910762823} a^{17} + \frac{7417408585300926214}{191625020136636387867} a^{15} - \frac{1897947182196454820315}{13222126389427910762823} a^{13} - \frac{1318568220689066059628}{13222126389427910762823} a^{11} + \frac{1485081508754775462682}{13222126389427910762823} a^{9} + \frac{675479033638037549593}{4407375463142636920941} a^{7} - \frac{1171934929997722413788}{13222126389427910762823} a^{5} - \frac{2732452161676229182828}{13222126389427910762823} a^{3} - \frac{330130655440485136625}{13222126389427910762823} a$

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:  $17$
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:  \( 4027410521370 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1738687177114624.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
89Data not computed