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Properties

Label 20.6.18513396751...5088.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $36.59$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-727, 0, 2930, 0, 16667, 0, 17481, 0, 4528, 0, -2239, 0, -1440, 0, -125, 0, 67, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 16*x^18 + 67*x^16 - 125*x^14 - 1440*x^12 - 2239*x^10 + 4528*x^8 + 17481*x^6 + 16667*x^4 + 2930*x^2 - 727)
 
gp: K = bnfinit(x^20 + 16*x^18 + 67*x^16 - 125*x^14 - 1440*x^12 - 2239*x^10 + 4528*x^8 + 17481*x^6 + 16667*x^4 + 2930*x^2 - 727, 1)
 

Normalized defining polynomial

\( x^{20} + 16 x^{18} + 67 x^{16} - 125 x^{14} - 1440 x^{12} - 2239 x^{10} + 4528 x^{8} + 17481 x^{6} + 16667 x^{4} + 2930 x^{2} - 727 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18513396751633196101360537305088=-\,2^{20}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.59$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{43} a^{16} - \frac{9}{43} a^{14} - \frac{16}{43} a^{12} - \frac{6}{43} a^{10} - \frac{17}{43} a^{8} - \frac{9}{43} a^{6} + \frac{13}{43} a^{4} + \frac{19}{43} a^{2} + \frac{19}{43}$, $\frac{1}{43} a^{17} - \frac{9}{43} a^{15} - \frac{16}{43} a^{13} - \frac{6}{43} a^{11} - \frac{17}{43} a^{9} - \frac{9}{43} a^{7} + \frac{13}{43} a^{5} + \frac{19}{43} a^{3} + \frac{19}{43} a$, $\frac{1}{6984296663529989} a^{18} - \frac{1416709670964}{162425503803023} a^{16} + \frac{2215450933835469}{6984296663529989} a^{14} - \frac{1055119363403956}{6984296663529989} a^{12} - \frac{1244632160266053}{6984296663529989} a^{10} - \frac{488736731866694}{6984296663529989} a^{8} - \frac{3138724256430274}{6984296663529989} a^{6} + \frac{3420107327747630}{6984296663529989} a^{4} + \frac{898730926014108}{6984296663529989} a^{2} + \frac{1519533371057747}{6984296663529989}$, $\frac{1}{6984296663529989} a^{19} - \frac{1416709670964}{162425503803023} a^{17} + \frac{2215450933835469}{6984296663529989} a^{15} - \frac{1055119363403956}{6984296663529989} a^{13} - \frac{1244632160266053}{6984296663529989} a^{11} - \frac{488736731866694}{6984296663529989} a^{9} - \frac{3138724256430274}{6984296663529989} a^{7} + \frac{3420107327747630}{6984296663529989} a^{5} + \frac{898730926014108}{6984296663529989} a^{3} + \frac{1519533371057747}{6984296663529989} a$

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:  $12$
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:  \( 67806641.1832 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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
Arithmetically equvalently 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 $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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$2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
11Data not computed
727Data not computed