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Properties

Label 20.0.11238525471...2096.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 41^{18}$
Root discriminant $56.56$
Ramified primes $2, 41$
Class number $1420$ (GRH)
Class group $[1420]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 385, 0, 4576, 0, 18297, 0, 33577, 0, 30961, 0, 14809, 0, 3825, 0, 532, 0, 37, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 37*x^18 + 532*x^16 + 3825*x^14 + 14809*x^12 + 30961*x^10 + 33577*x^8 + 18297*x^6 + 4576*x^4 + 385*x^2 + 1)
 
gp: K = bnfinit(x^20 + 37*x^18 + 532*x^16 + 3825*x^14 + 14809*x^12 + 30961*x^10 + 33577*x^8 + 18297*x^6 + 4576*x^4 + 385*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 37 x^{18} + 532 x^{16} + 3825 x^{14} + 14809 x^{12} + 30961 x^{10} + 33577 x^{8} + 18297 x^{6} + 4576 x^{4} + 385 x^{2} + 1 \)

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:  \(112385254718210608313247485941252096=2^{20}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.56$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(164=2^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(133,·)$, $\chi_{164}(139,·)$, $\chi_{164}(141,·)$, $\chi_{164}(81,·)$, $\chi_{164}(83,·)$, $\chi_{164}(23,·)$, $\chi_{164}(25,·)$, $\chi_{164}(31,·)$, $\chi_{164}(163,·)$, $\chi_{164}(37,·)$, $\chi_{164}(105,·)$, $\chi_{164}(107,·)$, $\chi_{164}(45,·)$, $\chi_{164}(113,·)$, $\chi_{164}(51,·)$, $\chi_{164}(119,·)$, $\chi_{164}(57,·)$, $\chi_{164}(59,·)$, $\chi_{164}(127,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{747} a^{16} - \frac{7}{249} a^{14} - \frac{29}{249} a^{12} - \frac{37}{249} a^{10} + \frac{43}{747} a^{8} + \frac{19}{249} a^{6} - \frac{55}{249} a^{4} - \frac{50}{249} a^{2} - \frac{98}{747}$, $\frac{1}{747} a^{17} - \frac{7}{249} a^{15} - \frac{29}{249} a^{13} - \frac{37}{249} a^{11} + \frac{43}{747} a^{9} + \frac{19}{249} a^{7} - \frac{55}{249} a^{5} - \frac{50}{249} a^{3} - \frac{98}{747} a$, $\frac{1}{91327473} a^{18} + \frac{21173}{91327473} a^{16} - \frac{3041435}{30442491} a^{14} + \frac{624610}{30442491} a^{12} - \frac{395102}{91327473} a^{10} - \frac{9602377}{91327473} a^{8} + \frac{11574431}{30442491} a^{6} + \frac{11469536}{30442491} a^{4} - \frac{10766726}{91327473} a^{2} - \frac{7900126}{91327473}$, $\frac{1}{91327473} a^{19} + \frac{21173}{91327473} a^{17} - \frac{3041435}{30442491} a^{15} + \frac{624610}{30442491} a^{13} - \frac{395102}{91327473} a^{11} - \frac{9602377}{91327473} a^{9} + \frac{11574431}{30442491} a^{7} + \frac{11469536}{30442491} a^{5} - \frac{10766726}{91327473} a^{3} - \frac{7900126}{91327473} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{1420}$, which has order $1420$ (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:  \( \frac{3581236}{91327473} a^{19} + \frac{132884525}{91327473} a^{17} + \frac{639634783}{30442491} a^{15} + \frac{4629305641}{30442491} a^{13} + \frac{54320312113}{91327473} a^{11} + \frac{115391121083}{91327473} a^{9} + \frac{42724490132}{30442491} a^{7} + \frac{23915907788}{30442491} a^{5} + \frac{18630636445}{91327473} a^{3} + \frac{1812944645}{91327473} a \) (order $4$)
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:  \( 5104264.63655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{41})\), 5.5.2825761.1, 10.0.335239100819416064.1, 10.10.327381934393961.1, 10.0.8176563434619904.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$