Properties

Label 20.0.56946837901...6801.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 17^{10}$
Root discriminant $10.91$
Ramified primes $7, 17$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 6*x^16 + 11*x^15 - 13*x^14 - 4*x^13 + 37*x^12 - 63*x^11 + 41*x^10 + 4*x^9 + 12*x^8 - 58*x^7 + 30*x^6 + 7*x^5 + 9*x^4 - 12*x^3 + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 6*x^16 + 11*x^15 - 13*x^14 - 4*x^13 + 37*x^12 - 63*x^11 + 41*x^10 + 4*x^9 + 12*x^8 - 58*x^7 + 30*x^6 + 7*x^5 + 9*x^4 - 12*x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -12, 9, 7, 30, -58, 12, 4, 41, -63, 37, -4, -13, 11, 6, -12, 8, -4, 1]);
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 12 x^{17} + 6 x^{16} + 11 x^{15} - 13 x^{14} - 4 x^{13} + 37 x^{12} - 63 x^{11} + 41 x^{10} + 4 x^{9} + 12 x^{8} - 58 x^{7} + 30 x^{6} + 7 x^{5} + 9 x^{4} - 12 x^{3} + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(569468379011812486801=7^{10}\cdot 17^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.91$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $20$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{18} - \frac{1}{26} a^{17} + \frac{1}{26} a^{16} - \frac{5}{26} a^{15} - \frac{1}{2} a^{14} + \frac{5}{26} a^{13} + \frac{1}{13} a^{12} - \frac{5}{26} a^{11} - \frac{6}{13} a^{10} + \frac{6}{13} a^{9} - \frac{5}{26} a^{8} - \frac{7}{26} a^{7} - \frac{1}{13} a^{6} + \frac{3}{26} a^{5} + \frac{4}{13} a^{4} - \frac{7}{26} a^{3} - \frac{5}{26} a^{2} - \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{642049175534} a^{19} - \frac{3783435379}{321024587767} a^{18} - \frac{67684022219}{642049175534} a^{17} + \frac{146314195525}{642049175534} a^{16} - \frac{63574974167}{321024587767} a^{15} + \frac{5717541913}{642049175534} a^{14} + \frac{26458292713}{642049175534} a^{13} + \frac{262972886}{24694199059} a^{12} + \frac{98609335257}{321024587767} a^{11} - \frac{202713022049}{642049175534} a^{10} - \frac{45179684889}{642049175534} a^{9} - \frac{11646478351}{49388398118} a^{8} - \frac{15444796061}{49388398118} a^{7} + \frac{18353610819}{642049175534} a^{6} + \frac{37673426969}{642049175534} a^{5} + \frac{13774315511}{49388398118} a^{4} - \frac{226202287847}{642049175534} a^{3} - \frac{199286226}{321024587767} a^{2} + \frac{262217054027}{642049175534} a + \frac{58351713835}{642049175534}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 77.3255035829 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}, \sqrt{17})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.1403737447.1 x5, 10.2.3409076657.1 x5

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

Sibling fields

Degree 10 siblings: 10.0.1403737447.1, 10.2.3409076657.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$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$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$