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Properties

Label 20.16.9773888551...1296.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{40}\cdot 31^{4}\cdot 557^{4}$
Root discriminant $28.15$
Ramified primes $2, 31, 557$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 34, 19, -640, -310, 3148, 322, -6870, 2165, 6682, -4955, -1716, 3223, -960, -434, 422, -150, 12, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 + 12*x^17 - 150*x^16 + 422*x^15 - 434*x^14 - 960*x^13 + 3223*x^12 - 1716*x^11 - 4955*x^10 + 6682*x^9 + 2165*x^8 - 6870*x^7 + 322*x^6 + 3148*x^5 - 310*x^4 - 640*x^3 + 19*x^2 + 34*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 + 12*x^17 - 150*x^16 + 422*x^15 - 434*x^14 - 960*x^13 + 3223*x^12 - 1716*x^11 - 4955*x^10 + 6682*x^9 + 2165*x^8 - 6870*x^7 + 322*x^6 + 3148*x^5 - 310*x^4 - 640*x^3 + 19*x^2 + 34*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} + 12 x^{17} - 150 x^{16} + 422 x^{15} - 434 x^{14} - 960 x^{13} + 3223 x^{12} - 1716 x^{11} - 4955 x^{10} + 6682 x^{9} + 2165 x^{8} - 6870 x^{7} + 322 x^{6} + 3148 x^{5} - 310 x^{4} - 640 x^{3} + 19 x^{2} + 34 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(97738885512935359526308151296=2^{40}\cdot 31^{4}\cdot 557^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.15$
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:  $2, 31, 557$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{59} a^{18} - \frac{20}{59} a^{17} - \frac{27}{59} a^{16} + \frac{18}{59} a^{15} + \frac{13}{59} a^{14} - \frac{26}{59} a^{13} + \frac{8}{59} a^{12} - \frac{18}{59} a^{11} - \frac{21}{59} a^{10} + \frac{8}{59} a^{9} - \frac{10}{59} a^{8} - \frac{21}{59} a^{7} + \frac{16}{59} a^{6} - \frac{15}{59} a^{5} + \frac{13}{59} a^{4} + \frac{10}{59} a^{3} - \frac{5}{59} a^{2} - \frac{8}{59} a + \frac{6}{59}$, $\frac{1}{45889855778459286730561} a^{19} - \frac{4977995957556053945}{777794165736598080179} a^{18} + \frac{15444879103364061213102}{45889855778459286730561} a^{17} - \frac{4685337863909508264319}{45889855778459286730561} a^{16} + \frac{19387778027313216436918}{45889855778459286730561} a^{15} - \frac{16729234279180479327077}{45889855778459286730561} a^{14} - \frac{8359635238973961943570}{45889855778459286730561} a^{13} + \frac{6535146189168262356428}{45889855778459286730561} a^{12} - \frac{16882502553693692962322}{45889855778459286730561} a^{11} - \frac{13404664299473830794860}{45889855778459286730561} a^{10} - \frac{20717668008012668297978}{45889855778459286730561} a^{9} + \frac{2379941376871471720725}{45889855778459286730561} a^{8} + \frac{8897634470609267678108}{45889855778459286730561} a^{7} - \frac{12401910362982946903997}{45889855778459286730561} a^{6} + \frac{13659301428674410940960}{45889855778459286730561} a^{5} - \frac{2065286061927186220451}{45889855778459286730561} a^{4} - \frac{16380325142764964792766}{45889855778459286730561} a^{3} + \frac{12369604193644502031652}{45889855778459286730561} a^{2} + \frac{5760289236509640525030}{45889855778459286730561} a - \frac{14170390877348622829011}{45889855778459286730561}$

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:  $17$
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:  \( 68652763.3501 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.138136.1, 10.8.305304871936.1, 10.8.156316094431232.1, 10.10.39079023607808.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
557Data not computed