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Properties

Label 20.20.4239038862...8576.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 401^{8}$
Root discriminant $53.87$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43823, 303198, -732298, 491158, 832863, -1531518, 269898, 1088072, -700797, -260128, 360010, -16828, -85710, 19738, 10188, -3820, -493, 318, -6, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 6*x^18 + 318*x^17 - 493*x^16 - 3820*x^15 + 10188*x^14 + 19738*x^13 - 85710*x^12 - 16828*x^11 + 360010*x^10 - 260128*x^9 - 700797*x^8 + 1088072*x^7 + 269898*x^6 - 1531518*x^5 + 832863*x^4 + 491158*x^3 - 732298*x^2 + 303198*x - 43823)
 
gp: K = bnfinit(x^20 - 10*x^19 - 6*x^18 + 318*x^17 - 493*x^16 - 3820*x^15 + 10188*x^14 + 19738*x^13 - 85710*x^12 - 16828*x^11 + 360010*x^10 - 260128*x^9 - 700797*x^8 + 1088072*x^7 + 269898*x^6 - 1531518*x^5 + 832863*x^4 + 491158*x^3 - 732298*x^2 + 303198*x - 43823, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 6 x^{18} + 318 x^{17} - 493 x^{16} - 3820 x^{15} + 10188 x^{14} + 19738 x^{13} - 85710 x^{12} - 16828 x^{11} + 360010 x^{10} - 260128 x^{9} - 700797 x^{8} + 1088072 x^{7} + 269898 x^{6} - 1531518 x^{5} + 832863 x^{4} + 491158 x^{3} - 732298 x^{2} + 303198 x - 43823 \)

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:  \(42390388623295510152536086938648576=2^{30}\cdot 3^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.87$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{10} + \frac{2}{9} a^{8} - \frac{4}{9} a^{7} + \frac{2}{9} a^{6} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{207} a^{17} - \frac{5}{207} a^{16} + \frac{5}{69} a^{15} - \frac{1}{207} a^{14} - \frac{8}{69} a^{12} + \frac{20}{207} a^{11} + \frac{9}{23} a^{10} + \frac{74}{207} a^{9} - \frac{4}{207} a^{8} - \frac{88}{207} a^{7} - \frac{62}{207} a^{6} - \frac{32}{207} a^{5} + \frac{1}{9} a^{4} - \frac{74}{207} a^{3} + \frac{31}{207} a^{2} + \frac{4}{207} a - \frac{2}{23}$, $\frac{1}{621} a^{18} + \frac{13}{621} a^{16} - \frac{41}{621} a^{15} - \frac{5}{621} a^{14} + \frac{91}{621} a^{13} - \frac{100}{621} a^{12} - \frac{95}{621} a^{11} - \frac{1}{23} a^{10} + \frac{11}{23} a^{9} - \frac{131}{621} a^{8} + \frac{32}{207} a^{7} - \frac{158}{621} a^{6} - \frac{61}{207} a^{5} - \frac{74}{621} a^{4} - \frac{17}{621} a^{3} - \frac{232}{621} a^{2} - \frac{113}{621} a - \frac{67}{621}$, $\frac{1}{3386245258416031983} a^{19} + \frac{150603305543722}{376249473157336887} a^{18} + \frac{503945606182495}{3386245258416031983} a^{17} - \frac{122547834880053323}{3386245258416031983} a^{16} + \frac{17740928023646692}{3386245258416031983} a^{15} + \frac{30765894409294552}{3386245258416031983} a^{14} - \frac{31496889965117680}{3386245258416031983} a^{13} + \frac{186520069817362363}{3386245258416031983} a^{12} - \frac{187875014940987595}{376249473157336887} a^{11} + \frac{51319294872956063}{376249473157336887} a^{10} + \frac{393257881098197950}{3386245258416031983} a^{9} + \frac{410275618826246243}{1128748419472010661} a^{8} + \frac{408824853387784132}{3386245258416031983} a^{7} + \frac{377573196541052102}{1128748419472010661} a^{6} + \frac{1090362372671817916}{3386245258416031983} a^{5} + \frac{654855572990190025}{3386245258416031983} a^{4} + \frac{742879527378996395}{3386245258416031983} a^{3} - \frac{269557958340592340}{3386245258416031983} a^{2} + \frac{276300143198330927}{3386245258416031983} a - \frac{29639401009434885}{125416491052445629}$

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:  \( 108189424779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{3}) \), 5.5.160801.1, 10.10.6434039469100032.1, 10.10.79432586038272.1, 10.10.2144679823033344.1

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 20 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed