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Properties

Label 20.0.42092092186...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{26}\cdot 7^{10}$
Root discriminant $21.44$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100, 0, -625, 0, 575, 0, 1125, 0, 1275, 0, 1090, 0, 450, 0, 85, 0, 15, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 15*x^16 + 85*x^14 + 450*x^12 + 1090*x^10 + 1275*x^8 + 1125*x^6 + 575*x^4 - 625*x^2 + 100)
 
gp: K = bnfinit(x^20 + 5*x^18 + 15*x^16 + 85*x^14 + 450*x^12 + 1090*x^10 + 1275*x^8 + 1125*x^6 + 575*x^4 - 625*x^2 + 100, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} + 15 x^{16} + 85 x^{14} + 450 x^{12} + 1090 x^{10} + 1275 x^{8} + 1125 x^{6} + 575 x^{4} - 625 x^{2} + 100 \)

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:  \(420920921862125396728515625=5^{26}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.44$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{40} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{3}{16} a^{6} + \frac{1}{16} a^{5} + \frac{5}{16} a^{3} - \frac{5}{16} a^{2} - \frac{1}{4}$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{11} - \frac{1}{80} a^{10} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{5}{16} a^{2} + \frac{1}{4}$, $\frac{1}{160} a^{14} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3} + \frac{5}{32} a^{2} - \frac{1}{8}$, $\frac{1}{320} a^{15} - \frac{1}{320} a^{14} - \frac{1}{80} a^{11} - \frac{1}{80} a^{10} + \frac{1}{32} a^{9} + \frac{3}{32} a^{8} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{17}{64} a^{3} - \frac{25}{64} a^{2} - \frac{5}{16} a - \frac{3}{16}$, $\frac{1}{640} a^{16} - \frac{1}{640} a^{14} - \frac{1}{160} a^{12} + \frac{3}{320} a^{10} + \frac{5}{64} a^{8} - \frac{1}{4} a^{6} - \frac{11}{128} a^{4} - \frac{45}{128} a^{2} - \frac{1}{2} a - \frac{3}{32}$, $\frac{1}{1280} a^{17} - \frac{1}{1280} a^{16} - \frac{1}{1280} a^{15} + \frac{1}{1280} a^{14} - \frac{1}{320} a^{13} + \frac{1}{320} a^{12} - \frac{1}{128} a^{11} - \frac{3}{640} a^{10} - \frac{3}{128} a^{9} + \frac{11}{128} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{59}{256} a^{5} - \frac{53}{256} a^{4} + \frac{35}{256} a^{3} - \frac{115}{256} a^{2} + \frac{29}{64} a + \frac{3}{64}$, $\frac{1}{440320} a^{18} + \frac{13}{220160} a^{16} + \frac{561}{440320} a^{14} + \frac{429}{220160} a^{12} - \frac{887}{110080} a^{10} + \frac{4767}{44032} a^{8} - \frac{8683}{88064} a^{6} + \frac{8013}{44032} a^{4} - \frac{1}{2} a^{3} - \frac{4587}{88064} a^{2} - \frac{1}{2} a - \frac{2097}{22016}$, $\frac{1}{880640} a^{19} - \frac{1}{880640} a^{18} + \frac{13}{440320} a^{17} - \frac{13}{440320} a^{16} + \frac{561}{880640} a^{15} - \frac{561}{880640} a^{14} + \frac{429}{440320} a^{13} - \frac{429}{440320} a^{12} + \frac{373}{44032} a^{11} + \frac{887}{220160} a^{10} - \frac{737}{88064} a^{9} - \frac{4767}{88064} a^{8} - \frac{8683}{176128} a^{7} - \frac{35349}{176128} a^{6} - \frac{19507}{88064} a^{5} + \frac{14003}{88064} a^{4} - \frac{37611}{176128} a^{3} + \frac{4587}{176128} a^{2} - \frac{13105}{44032} a + \frac{2097}{44032}$

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:  \( -1 \) (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:  \( 966410.388032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 5.1.765625.1 x5, 10.0.20516357421875.1, 10.0.4103271484375.1 x5, 10.2.2930908203125.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$